GIFT  OF 
Daughter  of  William 

Stuart  Smith.  U.S. Navy 


ENGINEERING  LIBRARY 


ALTERNATING  CURRENTS: 


AN  ANALYTICAL  AND  GRAPHICAL  TREATMENT 
FOR  STUDENTS  AND  ENGINEERS. 


BY 

FREDERICK  BEDELL,   PH.D., 

INSTRUCTOR  IN  PHYSICS,  CORNELL  UNIVERSITY, 

MEMB.   AM.   ASSOC.   FOR  THE  ADV.   OF  SCIENCE, 

ASSOC.  MEMB.  AM.  IN8T.  OF  ELECT.  ENG'RS, 

JUNIOR  MEMB.  AM.  SOC.  OF  MECH.  ENGR'S, 

AND 

ALBERT  GUSHING  CREHORE,  PH.D., 

INSTRUCTOR  IN  PHYSICS,   CORNELL  UNIVERSITY, 
ASSOC.   MEMB.   AM.   INST.   OF  ELEC.   ENG^RS. 


NEW  YORK: 

THE  W.  J.  JOHNSTON  COMPANY,  LIMITED, 
41  PARK  Row. 

LONDON  : 

WHITTAKER  &  CO.,  PATERNOSTER  SQUARE. 
1893. 


COPYRIGHT.  1893, 

BY 

F.  BEDELL  AND  A.  C.  CREHORE. 
^.W  rig/its  reserved. 


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ENGINEERING  LIBRARY 


PREFACE. 


THE  recent  advances  made  in  the  utilization  of  alter- 
nating currents  and  alternating  current  apparatus  of  all 
descriptions  have  been  of  such  importance  that  there 
are  now  many  interested  in  this  field  of  work  who  desire 
to  become  conversant  with  the  underlying  principles  of 
the  subject  in  order  that  they  may  become  better 
equipped  to  undertake  the  vast  engineering  problems 
which  are  constantly  arising.  In  its  newness,  the  theory 
of  alternating  currents  has  been  developed  this  way  and 
that,  added  to  here  and  there,  so  that  it  is  necessary 
for  one  to  stop  and  consider  the  basis  from  which  cer- 
tain conclusions  are  reached,  and  the  logicaL  sequence 
by  which  the  results  are  attained.  Although  many  of 
the  problems  which  arise  have  been  fully  treated  by 
various  writers,  the  solutions  have  as  a  rule  been  limited 
in  their  application  to  certain  special  cases,  and  have 
for  the  most  part  been  presented  in  a  fragmentary  man- 
ner. This  lack  of  a  clear  and  succinct  treatment,  suffi- 
ciently broad  to  be  general  in  its  application,  has  been 
strongly  felt,  and  it  is  in  order  to  meet  this  demand  for 
definite  information  in  regard  to  the  fundamental  prin- 
ciples governing  the  flow  of  variable  or  alternating  cur- 
rents that  this  work  is  now  presented  to  the  public. 

The  purpose  has  been  to  use  such  mathematical  terms 

and  analytical  methods  as  make  it  possible  for  the  dem- 

1 

888767 


2  PREFACE. 

onstrations  to  be  exact  and  rigorous,  and  at  the  same 
time  to  express  the  results  in  such  a  way  as  to  be  per- 
fectly intelligible  to  those  who  do  not  desire  to  follow 
the  methods  of  proof,  but  are  only  interested  in  the  con- 
clusions reached. 

There  are  some  to  whom  graphical  methods  appeal 
more  strongly  than  analytical  processes,  and  the  cases 
of  simple  circuits  have  accordingly  been  fully  treated  in 
both  ways.  The  problems  of  divided  circuits  and  net- 
works of  conductors  yield  the  more  readily  to  graphical 
treatment,  inasmuch  as  analytical  methods  necessarily 
become  cumbersome  and  involved  and  do  not  appeal 
directly  to  the  senses.  The  subject  is  therefore  capable 
of  two  natural  divisions,  the  analytical,  which  constitutes 
Part  I.,  and  the  graphical,  which  constitutes  Part  II. 

In  Part  I.  the  discussion  of  circuits  containing  re- 
sistance and  self-induction  only  is  first  taken  up,  and  the 
first  chapter  contains  the  elementary  principles  neces- 
sary for  the  establishment  of  the  equation  of  energy  for 
such  circuits.  This  equation  is  logically  developed 
from  the  experiments  of  Coulomb,  Faraday,  Joule,  and 
Ohm,  upon  which  depends  all  the  modern  science  of 
electricity.  The  treatment  is  based  upon  simple  ele- 
mentary ideas  and  is  complete  in  itself,  so  that  no  pre- 
vious knowledge  of  the  theory  of  electricity  and  mag- 
netism is  requisite.  Taking  the  equation  of  energy  as  a 
basis,  in  the  following  chapters  the  general  solution  for 
the  current  is  obtained,  after  which  the  various  particu- 
lar cases  are  taken  up,  in  which  the  electromotive  force 
is  assumed  to  vary  as  some  definite  function  of  the  time. 
The  solution  for  each  particular  case  is  derived  inde- 
pendently from  the  differential  equations  and  also  from 
the  general  integral  equation. 

Inasmuch  as  the  assumption  of  an  harmonic  electro- 
motive force  often  approximates  to  the  truth,  a  chapter 


PREFACE.  3 

has  been  devoted  to  the  discussion  of  harmouic  func- 
tions in  order  that  the  solutions  obtained  under  such  an 
assumption  may  be  the  more  clearly  understood. 

As  is  explained  in  the  introductory  chapter,  the  coeffi- 
cient of  self-induction  L  is  considered  constant,  whereas 
this  is  only  strictly  true  if  the  permeability  of  the  sur- 
rounding medium  is  also  constant.  That  this  assump- 
tion is  nearly  correct  is  readily  seen  by  noting  the  curves 
of  magnetization  for  various  commercial  irons  given  by 
Prof.  Ewing,  and  by  Mr.  Steinmetz  and  Mr.  M.  E. 
Thompson  in  this  country,  for  it  is  not  until  a  higher 
degree  of  magnetization  is  reached  than  is  ordinarily 
met  with  in  actual  practice,  that  these  curves  deviate 
materially  from  a  straight  line. 

After  the  completion  of  the  treatment  of  circuits  con- 
taining resistance  and  self-induction,  the  discussion  of  cir- 
cuits containing  resistance  and  capacity  is  taken  up  and 
developed  in  a  similar  manner  from  elementary  princi- 
ples. From  the  simple  ideas  of  static  charge,  the  mean- 
ing of  potential  and  work  is  shown,  leading  up  to  the 
derivation  of  the  equation  of  energy  and  electromotive 
forces  for  a  circuit  containing  a  condenser.  Following 
the  same  plan  as  in  the  treatment  of  circuits  containing 
self-induction,  the  general  solution  is  first  obtained,  and 
we  thus  have  the  expression  for  the  current  and  charge 
at  any  time  for  any  impressed  electromotive  force  what- 
soever. Particular  electromotive  forces  are  then  as- 
sumed, and  the  solutions  for  these  cases  are  obtained 
from  the.  general  integral  equation,  and  also  independ- 
ently by  particular  solutions. 

The  general  case  of  circuits  containing  resistance,  self- 
induction,  and  capacity  is  next  taken  up,  and  the  same 
order  of  treatment  is  followed  as  in  the  discussion  of 
circuits  containing  resistance  and  self  induction  only, 
and  resistance  and  capacity  only.  Now  that  the  con- 


4  PREFACE. 

denser,  as  well  as  its  older  brother,  the  transformer,  is 
being  applied  to  practical  uses,  the  question  of  the  action 
of  a  condenser  in  a  circuit  with  self-induction  becomes 
an  important  one,  and  the  discussion  of  this  case  is  given 
at  length,  the  same  method  of  giving  particular  cases 
after  the  general  solution  being  followed  as  before.  The 
case  of  oscillatory  and  non-oscillatory  charge  is  treated 
at  length  as  well  as  the  corresponding  case  of  discharge. 
In  order  that  the  effects  caused  by  the  variation  of  the 
constants  of  a  circuit  may  be  clearly  understood,  curves 
are  drawn  showing  these  effects  for  certain  particular 
cases.  The  nature  of  the  flow  of  current  immediately 
after  making  a  circuit  is  then  investigated,  and  the  re- 
sults shown  by  plotting  the  instantaneous  values  for  a 
particular  case.  The  neutralizing  effects  of  self-induc- 
tion and  capacity  are  next  discussed,  and  the  necessary 
conditions  ascertained  under  which  not  only  the  instan- 
taneous values  of  the  current  will  be  the  same  as  though 
the  self-induction  and  capacity  were  absent,  but  likewise 
the  thermic  and  dynamometric  effects. 

The  first  part  closes  with  an  investigation  of  the 
nature  of  wave  propagation  in  a  conductor  possessing 
self-induction  and  distributed  capacity,  a  subject  which 
assumes  importance  in  submarine  cables  and  in  ex- 
tended telephone  circuits. 

The  results  obtained  by  analytical  processes  too  often 
fail  to  carry  their  full  significance  while  in  symbolic 
form,  and  for  this  reason  it  has  been  found  advisable  to 
give  applications  to  concrete  cases,  and  to  draw  curves 
illustrating  the  points  involved.  In  order  that  the  full 
significance  of  the  results  may  be  grasped,  the  values  of 
the  quantities  used  in  these  numerical  examples  have  in 
all  cases  been  given,,  so  that  the  curves  plotted  show  not 
only  the  general  nature  of  the  relations  between  the 
various  quantities,  but  also  the  value  of  these  quantities 


PREFACE.  5 

in  the  particular  cases  assumed.  The  advantage  of  this 
is  especially  shown  in  the  discussion  of  the  effects  of  the 
variation  of  the  constants  in  a  circuit  containing  resist- 
ance, self-induction  and  capacity,  for  it  is  the  illustra- 
tions which  here  bring  out  the  true  significance  of  the 
effects. 

In  Part  II.  the  same  order  is  followed  as  in  Part  I. 
The  graphical  method  of  treating  problems  of  simple 
circuits  containing  resistance  and  self-induction  is  first 
fully  established  from  the  analytical  results  obtained  in 
Part  I.,  and  is  then  extended  to  problems  arising  in  the 
case  of  combination  circuits.  Problems  arising  in  the 
case  of  simple  and  combination  circuits  containing  re- 
sistance and  capacity  but  no  self-induction  are  then 
solved,  and  finally  the  general  case  of  circuits  containing 
resistance,  self-induction  and  capacity  is  taken  up,  and 
the  graphical  solutions  given  for  series,  parallel  and 
combined  circuits. 

The  graphical  methods  are  rigorously  proved  by  the 
analytical  solutions  obtained  in  the  earlier  part  of  the 
book,  but  the  development  is  such  that  those  who  do 
not  follow  through  the  analytical  proof  may  readily 
apply  these  graphical  methods  to  the  solution  of  practi- 
cal problems. 

In  order  to  avoid  ambiguity,  the  same  symbols  are 
used  throughout  with  the  same  signification,  and  a  list 
of  symbols  used,  together  with  their  meanings,  is  given 
in  an  appendix. 

There  have  been  many  valuable  papers  on  subjects 
relating  to  alternating  currents,  among  others  those  by 
Dr.  Duncan  and  Prof.  Ryan  in  this  country,  and  by  Prof. 
Ayrton,  Dr.  Sumpner,  Dr.  Fleming,  and  Mr.  Blakesley  in 
England,  and  the  electrical  public  has  gained  much  in- 
formation from  the  excellent  works  of  the  last  two 
writers.  The  subject  has  not,  however,  been  hitherto 


6  PREFACE. 

developed  in  the  way  followed  in  the  succeeding  pages, 
and  it  is  in  order  to  meet  the  demand  for  a  logical  treat- 
ment of  the  theory  of  alternating  currents  that  this  book 
has  been  prepared. 

Much  of  the  matter  here  contained  has  already  been, 
given  by  the  writers  in  various  papers,  some  of  which 
originally  appeared  as  a  series  of  articles  in  the  Electrical 
World,  and  others  in  the  London  Electrician,  the  Philo- 
sophical Magazine,  the  American  Journal  of  Science,  and 
the  Transactions  of  the  American  Institute  of  Electrical  En- 
gineers. We  have  been  permitted  to  use  some  of  the 
cuts  from  the  latter,  for  which  courtesy  we  desire  to 
extend  our  thanks. 

The  matter  contained  in  the  second  part  now  appears 
for  the  first  time,  with  the  exception  of  the  method  for 
obtaining  the  equivalent  resistance,  self-induction  and 
capacity  of  parallel  circuits,  which  was  first  given  in  the 
Philosophical  Magazine. 

In  all  cases  these  papers  have  been  carefully  revised 
and  rewritten,  and  in  many  cases  amplified  to  suit  the 
requirements  of  the  book. 

CORNELL  UNIVERSITY,  ITHACA,  N.  Y., 
August,  1892. 


CONTENTS. 


PART  I.    ANALYTICAL  TREATMENT. 

CHAPTER  I. 

INTRODUCTORY  TO  CIRCUITS  CONTAINING  RESISTANCE  AND  SELF- 
INDUCTION. 

Magnet.  Lines  of  force.  Field  of  force.  Pole.  North  pole.  Like  poles 
repel,  unlike  attract.  Unit  pole.  Law  of  attraction.  Intensity  of 
a  field  of  force.  Uniform  field.  Unit  line  of  force.  Unit  pole 
has  kit  lines  of  force.  Induction.  Current  develops  a  field.  Unit 
current.  Number  of  lines  proportional  to  current.  Self-induction. 
E.  M.  F.  Ohm's  law.  Quantity.  Quantity  definite  for  definite  change 
in  lines.  Joule's  law.  Energy  dissipated  in  heat.  Total  energy  im- 
parted to  a  circuit.  Energy  expended  in  field.  Equation  of  energy. 
Equation  of  E.  M .  F.  's.  Pages  17-31. 

CHAPTER  II. 
ON  HARMONIC  FUNCTIONS. 

Harmonic  E.  M.  F.  often  assumed.  Simple  harmonic  motion.  Ampli- 
tude. Period.  Angular  velocity.  Frequency.  Epoch.  Phase.  Lag. 
Advance.  Graphical  representation  of  simple  harmonic  functions. 
Average  value  of  ordinates  of  sine-curve.  Value  of  mean  square  of 
ordinates  of  sine-curve.  Periodic  functions  composed  of  several  simple 
sine-functions  of  like  period, — of  unlike  period.  Fourier's  theorem. 

Pages  33-41. 
CHAPTER  III. 

CIRCUITS  CONTAINING  RESISTANCE  AND  SELF-INDUCTION. 

Equations  of  energy  and  E.  M.  F.'s.    Criterion  of  integrability.    General 
solution  when  e  =f(t). 

Case  I.  E.  M.  F.  suddenly  Removed.  Solution  from  differential  equa- 
tion,— from  general  solution.  Geometric  construction  of 
logarithmic  curve. 

Case  II.  E.  M.  F.  suddenly  Introduced.  Solution  from  differential 
equation, — from  general  solution. 

7 


8  CONTENTS. 

Case  III.  Simple  Harmonic  E.  M.  F.     Solution  from  general  equation. 

Impedance.     Lag.     Effect  of  exponential  term  at  "  make." 
Case  IV.  Any  Periodic  E.  M.  F.     Sum  of  two  sine-functions.     Sum  of 

any  number  of  sine-functions.  Pages  42-59. 

CHAPTER  IV. 
INTRODUCTORY  TO  CIRCUITS  CONTAINING  RESISTANCE  AND  CAPACITY. 

Plan  to  be  followed.  Charge.  Law  of  force.  Unit  charge.  Work  in 
moviug  a  charge.  Potential.  Capacity.  Energy  of  charge.  Con- 
denser,—energy  of  aud  capacity  of.  Capacity  of  parallel  plates  ;  of 
continuous  conductor.  Equation  of  energy,  in  terms  of  i\  in  terms 
of  q.  Equation  of  E.  M.  F.'s.  Pages  60-69. 

CHAPTER  V. 
CIRCUITS  CONTAINING  RESISTANCE  AND  CAPACITY. 

Equation  of  E.  M.  F.'s.  Differential  equation  in  linear  form.  Criterion  of 
integrability.  General  solution  when  e  =f(t). 

Case  I.  Discharge.  Quantity  and  current  from  general  solution, — from 
differential  equations. 

Case  II.  Charge.  Quantity  and  current  from  general  solution,— from  dif- 
ferential equations. 

Case  III.  Simple  harmonic  E.  M.  F.  Quantity  and  current  from  general 
solution.  Discussion. 

Case  IV.  Any  periodic  E.  M.  F.  Pages  70-80. 

CHAPTER  VI. 

CIRCUITS  CONTAINING  RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY. 
GENERAL  SOLUTION. 

Equation  of  energy  in  terms  of  e,  i,  and  t ;  in  terms  of  e,  q,  and  t.  Equa- 
tion of  E.  M.  F.'s  in  terms  of  e,  i,  and  t ;  in  terms  of  e,  q,  and  t. 
Equations  transformed  for  solving  in  terms  of  i  aud  t ;  in  terms  of 
q  and  t.  Complete  solution  for  i  in  terms  of  t  ;  complete  solution  for  q 
in  terms  of  t.  Four  cases  will  be  considered:  I.  e  =f(t)  =  0;  II. 
e=f(t)  =  E;  Ill.e=f(t)  =  E  sin  <vt;  IV.  e  =f(t)  =  2  E  sin  (&&?<,  +  &) 

Pages  81-89. 

CHAPTER  VII. 

CIRCUITS  CONTAINING  RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY. 
CASE  I.   DISCHARGE. 

Integral  and  differential  equations  when  e  =f(t)  =  0.  Sir  Win.  Thomson's 
solution,  i  equation  with  value  of  T  replaced.  Three  forms  of  i  aud  q 
equations.  To  transform  the  ^-equation  to  a  real  form  when  R?G  is 
less  than  4L.  To  derive  the  solutions  from  the  differential  equations 
when 


CONTENTS.  9 

Non-oscillatory  Discharge. 

Determination  of  constants.  Complete  solution.  Value  of  T  re- 
placed. Current  and  charge  curves  for  a  particular  circuit.  Time  of 
maximum  current.  Equation  (125)  applied  to  a  circuit  containing  resist- 
ance and  self-induction  only,  and  to  a  circuit  containing  resistance 
and  capacity  only. 

Oscillatory  Discharge.  » 

Determination  of  constants.  Complete  solution  for  i  and  q.  Current 
and  charge  curves  for  a  particular  circuit. 

Discharge  of  Condenser  when  R2C  =  4Z. 

Determination  of  constants.  Complete  solutions  for  i  and  q.  Figure 
showing  method  of  constructing  the  current  and  charge  curves.  Curves 
for  i  and  q  in  a  particular  circuit.  Pages  90-111. 

CHAPTER  VIII. 

CIRCUITS  CONTAINING  RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY. 
CASE  II.  CHARGE. 

Differential  equations  with  e=f(t)  =  E.  Solution  of  these  equations. 
Solution  from  the  general  integral  equation.  Three  forms  of  i  and  q 
equations. 

Non  oscillatory  Charging. 

Determination  of  constants.  Complete  solutions  for  *  and  q  with 
constants  determined  Curves  for  i  and  q  in  a  particular  circuit.  Equa- 
tion (101)  applied  to  a  circuit  containing  resistance  and  self-induction 
only  ;  also  to  a  circuit  containing  resistance  and  capacity  only. 

Oscillatory  Charging. 

Determination  of  constants  Complete  solutions  for  i  and  q  with 
constants  determined.  Curves  for  i  and  q  in  a  particular  circuit. 


Charge  of  the  Condenser  when 

Determination  of  constants.     Complete  solutions  for  i  and  q  with 
constants  determined.     Curves  for  i  and  q  in  a  particular  circuit. 

Pages  11  2-1  23. 
CHAPTER  IX. 

CIRCUITS  CONTAINING  RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY. 
CASE  III.   SOLUTION  AND  DISCUSSION  FOR  HARMONIC  E.  M.  F. 

To  find  from  the  general  solutions  the  particular  equations  in  the  case  of  an 
harmonic  E.  M.  F.  Complete  solutions  for  i  and  q.  These  same  solu- 
tions obtained  directly  from  the  differential  equations. 

Discussion  of  Case  III.     Harmonic  E.  M.  F. 

The  impediment.     Case  A.  Circuits  containing  resistance  and  self- 
induction  only.     Case  B.  Circuits  containing  resistance  and  capacity 


10  CONTENTS. 

only.     Case  C.  Circuits  containing  resistance  only.     Case  D.  Circuits 
containing  capacity  only. 

Effects  of  Varying  the  Constants  of  a  Circuit. 

First.  Electromotive  force  varied.  Second.  Resistance  varied, 
Third.  Coefficient  of  self-induction  varied.  Fourth.  Capacity  varied. 
Fifth.  The  frequency  varied. 

The  energy  expended  per  second  upon  a  circuit  in  which  an  har- 
monic circuit  is  flowing.  Pages  124-143. 

CHAPTER  X. 

CIRCUITS  CONTAINING  RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY. 

CASE  III.  (CONTINUED.)    CURRENT  AT  THE  "MAKE"  FOR  AN 

HARMONIC  E.  M.  F. 

Complete  equations  for  i  and  q  with  the  complementary  function  in 
the  oscillatory  form.  To  determine  the  constants  A'  and  <£'.  To 
determine  the  constants  A  and  <£.  Complete  equation  for  i  with 
constants  determined.  Examples  to  explain  the  general  equation 
in  case  of  particular  circuits.  Curves  showing  the  current  at  the  make 
for  a  particular  circuit.  The  phase  at  which  the  E.  M.  F.  should  be 
introduced  to  make  the  oscillation  a  maximum.  Pages  144-155. 

CHAPTER  XI. 

CIRCUITS  CONTAINING  RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY. 
CASE  IV.    ANY  PERIODIC  E.  M.  F. 

Fourier's  theorem.  General  equations  for  i  and  q  with  any  periodic 
E.  M.  F.  If  the  self-induction  and  capacity  neutralize  each  other 
at  every  point  of  time  and  the  current  is  therefore  the  same  as  if 
both  self-induction  and  capacity  were  absent,  the  impressed  E.  IvI.  F. 
must  be  a  simple  harmonic  E.  M.  F.  If  the  heating  effect,  or  any 
effect  which  depends  upon  JPdt,  in  a  circuit,  is  the  same  when  the  self- 
induction  and  capacity  are  present  as  it  is  when  they  are  absent,  the 
impressed  E.  M.  F.  must  be  a  simple  harmonic  E.  M.  F.  Various 
types  of  current  curves.  When  curves  are  not  symmetrical,  although 
the  quantity  flowing  in  the  positive  direction  is  equ;il  to  the  quantity 
in  the  negative  direction,  yet  theffrdt  effect  will  generally  be  different 
in  these  two  directions.  Illustration  from  a  particular  curve.  Alter- 
nating-current arc-light  carbons.  Pages  156-175. 

CHAPTER  XII. 

CIRCUITS  CONTAINING  DISTRIBUTED  CAPACITY  AND  SELF-INDUCTION. 
GENERAL  SOLUTION. 

Derivation  of  the  differential  equation  for  circuits  containing  distributed 
capacity  only.  This  equation  extended  so  as  to  represent  a  particu- 
lar case  of  distributed  capacity  and  self-induction.  Differential 


CONTENTS.  11 

equation  for  E.  M  F.  is  of  the  same  form  as  that  for  current.  The 
general  solutions  of  the  differential  equations.  Particular  assumption 
of  harmonic  E.  M.  F.  Constants  of  the  general  equation  determined 
under  this  assumption;  first,  from  the  exponential  solution;  second, 
from  the  sine  solution.  Current  determined  from  the  E.  M.  F. 
equation.  Pages  176-192. 

CHAPTER  XIII. 

CIRCUITS  CONTAINING  DISTRIBUTED  CAPACITY  AND  SELF-INDUCTION. 
DISCUSSION. 

Circuits  with  no  self-induction.  Particular  form  of  e  and  i  equations. 
Nature  of  waves.  Rate  of  propagation.  Wave-length.  Decreasing 
amplitude.  Rate  of  decay  with  distance;  with  time. 

Circuits  icith  self-induction.  Phase  difference.  Rate  of  propagation. 
Diminishing  amplitude.  Rate  of  decay.  Limitations  of  the  tele- 
phone. 

Wave  propagation  in  Closed  Circuits 

Positive  and  negative  waves  travel  around  the  circuit  until  they 
vanish.  Resultant  effect.  Potential  zero  at  middle  point  of  the  cable. 
Expression  for  potential  simplified  if  the  length  of  the  cable  is 
a  multiple  of  a  wave-length.  Same  results  may  be  applied  to  the 
current  equation.  Pages  193-207. 


PART   II.— GRAPHICAL   TREATMENT. 


CHAPTER  XIV. 

INTRODUCTORY  TO  PART  II.  AND  TO  CIRCUITS  CONTAINING  RESISTANCE 
AND  SELF-INDUCTION. 

Introductory.  Analytical  solutions  of  Part  I.  for  simple  circuits  extended 
to  compound  circuits  by  graphical  method.  Arrangement  of  Part  II. 
Graphical  representation  of  simple  harmonic  E.  M.  F.  's.  Graphical  rep- 
resentation-of  the  sum  of  simple  harmonic  E  M.  F.'s  of  same  period. 
Triangle  of  E.  M.  F.'s  for  a  single  circuit  containing  resistance  and 
self-induction.  Impressed  E.  M.  F.  Effective  E.  M.  F.  Counter 
E.  M  F.  of  self-induction.  Direction  shown  from  differential  equation. 
Graphical  representation.  Methods  to  be  used  and  symbols  adopted 
in  the  graphical  treatment  of  problems.  First  method  (the  one  used 
throughout  this  book),  employing  E.  M.  F.  necessary  to  overcome  self- 
iiiduction.  Second  method,  employing  E.  M.  F.  of  self-induction. 
System  of  lettering  and  conventions  adopted  in  graphical  construction. 

Pages  211-221. 


12  CONTENTS. 

CHAPTER  XV. 

PROBLEMS  WITH  CIRCUITS  CONTAINING  RESISTANCE  AND  SELF-INDUCTION. 
SERIES  CIRCUITS  AND  DIVIDED  CIRCUITS. 

Prob.       L  Effects  of  the  Variation  of  the  Constants  R  and  L  in  a  Series 

Circuit.     R  varied.     L  varied. 
Prob.     II.  Series  Circuit.     Current  given. 
Prob.    III.  Series  Circuit.     Impressed  E.  M.  F.  given. 
Prob.  Ilia.  Measurements  on  a  Series  Circuit. 
Prob.    IV.  Divided  Circuit.     Two  Branches.     Impressed  E.  M.  F.  given. 

Equivalent  Resistance  and  Self-induction  defined. 
Prob.      V.  Divided    Circuit.      Any  Number    of    Branches.      Impressed 

E.  M.  F.  given.     Equivalent  Resistance  and  Self-induction 

obtained  for  Parallel  Circuits. 
Prob.    VI.  Divided    Circuit.     Current    given.      First    Method:    Entirely 

Graphical.     Second    Method:   Solution    by  Equivalent  R 

and  L. 
Prob.  VII.  Effects  of  the  Variation  of  the  Constants  R  and  L  in  a  Divided 

Circuit  of  Two  Branches.     R  varied.     L  varied.     Limiting 

Cases.     Constant    Potential   Example.     Constant    Current 

Example.  Pages  223-247. 


CHAPTER  XVI. 

PROBLEMS  WITH  CIRCUITS  CONTAINING  RESISTANCE  AND  SELF-INDUCTION. 
COMBINATION  CIRCUITS. 

Prob.  VIII.  Series    and    Parallel    Circuits.      Impressed  E.  M.  F.  given. 

Solution  by  Equivalent  R  and  Z. 
Prob.     IX.  Series  and  Parallel   Circuits.     Current  given.     Solution    by 

Equivalent  R  and  L. 

Prob.       X.  Extension  of  Problems  VIII.  and  IX. 
Prob.     XL  Series  and  Parallel  Circuits.     Entirely  Graphical  Solution. 
Prob.    XII.  Multiple-arc  Arrangement.  Pages  248-259. 


CHAPTER  XVII. 

PROBLEMS  WITH  CIRCUITS  CONTAINING  RESISTANCE  AND  SELF-INDUCTION. 
MORE  THAN  ONE  SOURCE  OF  ELECTROMOTIVE  FORCE. 

Prob.  XIII.  Electromotive  Forces  in  Series. 

Prob.  XIV.  Direction  of  Rotation  of  E.  M.  F.  Vectors. 

Prob.    XV.  Electromotive  Forces  in  Parallel. 

Prob.  XVI.  Electromotive  Forces  having  Different  Periods. 


CONTENTS.  13 

CHAPTER  XVIII. 
INTRODUCTORY  TO  CIRCUITS  CONTAINING  RESISTANCE  AND  CAPACITY. 

Problems  with  R  and  G  analytically  aud  graphically  analogous  to  prob- 
lems with  R  aud  L.  Triangle  of  E.  M.  F.'s  for  a  single  circuit 
containing  resistance  and  capacity.  Impressed  E.  M.  F.  Effec- 
tive E.  M.  F.  Condenser  E.  M.  F.  Direction  shown  from  differ- 
ential equations.  Graphical  representation.  Two  methods  used. 
First  method  (the  one  used  throughout  this  book),  employing  E.  M.  F. 
necessary. to  overcome  that  of  condenser.  Second  method,  employing 
E.  M.  F.  of  condenser.  Further  identification  of  analytical  and 
graphical  relations.  Mechanical  analogue.  Pages  267-273. 

CHAPTER  XIX. 

PROBLEMS  WITH  CIRCUITS  CONTAINING  RESISTANCE  AND  CAPACITY. 

Prob.      XVII.  Effects  of  the  Variation  of  the  Constants  R  and  C  in  a 

Series  Circuit.     .K  varied.     C  varied. 
Prob.    XVIII.  Series  Circuit.     Current  given.     Equivalent  R  and  C  in 

Series. 

Prob.       XIX.  Series  Circuit.     Impressed  E.  M.  F.  given. 
Prob.         XX.  Divided    Circuit.     Two    Branches.     Impressed    E.  M.  F. 

given.     Equivalent  R  and  C  for  Parallel  Circuit. 
Prob.       XXI.  Divided  Circuit.     Any  Number  of  Branches.     Impressed 

E.  M.  F.   given.     Equivalent   R  and    G  obtained   for 

Parallel  Circuits. 
Prob.     XXII.  Divided  Circuit.    Current  given.    First   Method:  Entirely 

Graphical.     Second  Method:  Solution  by  Equivalent  R 

and  C. 
Prob.    XXIII.  Effects  of  the  Variation  of  the  Constants  R  and  G  in  a 

Divided  Circuit  of  Two  Branches. 
Prob.    XXIV.  Series  and  Parallel  Circuits.     Impressed  E.  M.  F.  given. 

Solution  by  Equivalent  R  and  C. 
Prob.     XXV.  Series  and  Parallel  Circuits.     Current  given.     Solution  by 

Equivalent  R  and  C. 

Prob.    XXVI.  Series  and  Parallel  Circuits.     Entirely  Graphical  Solution. 
Prob.  XXVII.  Multiple-arc  Arrangement.  Pages  274-291. 

CHAPTER  XX. 

CIRCUITS  CONTAINING  RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY. 

Introductory.  Graphical  methods  for  circuits  with  R,  L,  and  C  based 
upon  "graphical  methods  for  circuits  with  R  and  L,  and  R  and  C. 
Diagram  of  four  E.  M.  F.'s.  Triangle  of  E.  M.  F.'s.  Method  con- 
sistent with  analytical  results  obtained  for  circuits  with  R,  L,  and  G. 


14    .  CONTENTS. 

Capacity  or  s'elf-inductiou  which  is  equivalent  to  a  combination  of 
capacity  aiid  self-induction. 
Prob.  XXVIII.  Effects  of  the  Variation  of  the  Constants  in  Series  Circuit. 

21,  L,  C,  and  GO  varied. 
Prob.      XXIX.  Series  Circuit.     Current  given.     Equivalent  R,  L,  and  C 

of  Series  Circuit. 

Prob.        XXX.  Series  Circuit.     Impressed  E.  M.  F.  given. 
1'rob.      XXXI.  Divided  Circuit.     Impressed  E.  M.  F.  given.     Equivalent 

M,  L,  and  C  of  Parallel  Circuits. 
Prob.    XXXII.  Example    of    a    Divided    Circuit.     Impressed   E.  M.  F. 

given. 

Prob.  XXXIII.  Divided  Circuit.     Current  given. 
Prob.  XXXIV.  Series  and  Parallel  Combinations  of  Circuits. 

Pages  292-311, 

APPENDIX  A. 

Relation  between  Practical  and  C.  G.  S.  Units.  Page  312. 


APPENDIX  B. 

Some  Mechanical  and  Electrical  Analogies."  Pages  313-315. 

APPENDIX  C. 

System  of  Notation  adopted.  Pages  316-318. 

INDEX.  Pages  319-325. 


PART  I. 

ANALYTICAL   TREATMENT. 


CHAPTER  I. 

INTRODUCTORY    TO    CIRCUITS    CONTAINING   RESISTANCE 
AND  SELF  INDUCTION. 

€ONTENTS: — Magnet.  Lines  of  force.  Field  of  force.  Pole.  North  pole. 
Like  poles  repel,  unlike  attract.  Unit  pole.  Law  of  attraction.  In- 
tensity of  a  field  of  force.  Uniform  field.  Unit  line  of  force.  Unit 
pole  lias  4:7t  lines  of  force.  Induction.  Current  develops  a  field.  Unit 
current.  Number  of  lines  proportional  to  current.  Self-induction. 
E.  M.  Fc  Ohm's  law.  Quantity.  Quantity  definite  for  definite  change 
in  lines.  Joule's  law.  Energy  dissipated  in  heat.  Total  energy  im- 
parted to  a  circuit.  Energy  expended  in  field.  Equation  of  energy. 
Equation  of  E.  M.  F.'s. 

IN  order  that  circuits  containing  resistance  and  self- 
induction  may  be  properly  discussed,  a  brief  review  will 
first  be  given  of  the  elementary  theory  of  magnetism,  the 
nature  of  the  magnetic  field,  and  the  relation  between  a 
current  of  electricity  and  magnetism.  Those  well-known 
elements  of  the  subject  will  be  presented  which  enable  us 
to  obtain  expressions  for  the  energy  imparted  to  a  circuit, 
the  energy  dissipated  in  heat,  and  the  energy  expended  in 
the  magnetic  iield,  and  finally  to  establish  the  equation  of 
energy  and  the  equation  of  electromotive  forces  for  circuits 
with  resistance  and  self-induction. 

If  a  needle  is  magnetized  uniformly  in  the  direction  of 
its  length  and  placed  in  iron  filings,  the  filings  are  at- 
tracted to  the  ends  of  the  needle  and  become  attached 
thereto  in  clusters.  The  attractive  power  of  the  magnet- 

17 


18  INTRODUCTORY  TO   CIRCUITS 

ized  needle  is  apparently  concentrated  at  the  ends,  which 
^are \called  -"$olfs.\  ?The  filings  in  the  space  around  the 
magnet  fend  to  garths  *v  in  lines,  called  lines  of  force,  extend- 
.in^fr^^on^tpoj^^f  ctlie  needle  to  the  other.  Thus  the 
magnet  is  seen  to  be  surrounded  by  afield  of  force,  in  which 
the  lines  indicate  the  direction  of  the  force  at  any  point  of 
the  field.  When  a  compass-needle  is  placed  in  the  field,  it 
always  assumes  a  definite  position,  tangent  to  the  line  of 
force  passing  through  that  point.  The  earth  acts  like  a 
huge  magnet,  producing  a  magnetic  field  in  which  the 
lines  of  force  have  a  direction  nearly  north  and  south.  -A 
magnetized  needle  freely  suspended  in  the  earth's  magnetic 
field  assumes  a  definite  position  tangent  to  the  earth's  lines 
of  force.  This  position  is  usually  nearly  in  the  geographical 
meridian,  the  magnet  having  one  pole  toward  the  north 
and  the  other  toward  the  south.  The  pole  that  is  toward 
the  north  is  called  the  positive  pole,  marked  -j-  ;  and  the 
opposite  pole  the  negative,  marked  — .  When  magnetic 
poles  are  brought  near  one  another  there  is  found  to  be 
either  an  attraction  or  a  repulsion  between  them,  and  two 
poles  which  have  the  same  sign  tend  to  repel  one  another, 
while  two  poles  of  opposite  sign  tend  to  attract  one  another. 
The  definition  of  a  unit  magnetic  pole  would  therefore 
naturally  be  :  a  magnetic  pole  which  exerts  a  force  of  one 
dyne*  upon  another  equal  pole  at  a  distance  of  one  centi- 

*  Our  knowledge  of  the  physical  universe  is  obtained  from  our  per- 
ception of  matter  in  its  relations  to  time  and  space;  and  physical  measure- 
ments are,  accordingly,  measurements  of  mass,  length,  and  time.  Any 
quantity  can  be  expressed  in  terms  of  these  three,  and  the  units  in  which 
the  quantity  is  measured  can  be  expressed  in  the  three  fundamental  units 
of  length,  mass,  and  time.  The  fundamental  units  commonly  used  to 
measure  length,  mass,  and  time  are  the  centimetre,  gramme,  and  second; 
these  are  arbitrarily  selected,  and  give  rise  to  the  C.  G.  S.  system  of  units. 
All  other  units  are  readily  obtained  from  these  and  are  called  derived  units. 
The  velocity  of  a  body  moving  uniformly  is  the  space  passed  over  in  a  unit 
time.  For  a  body  having  a  variable  motion,  the  velocity  is  equal  to  an 


CONTAINING  RESISTANCE  AND  SELF  INDUCTION.      19 

metre.  Such  a  magnetic  pole  as  tliis  just  defined  forms  the 
foundation  upon  which  is  based  the  whole  system  of  electro- 
magnetic units,  those  of  current,  electromotive  force,  etc.; 
and  it  therefore  deserves  attention. 

This  definition  depends  upon  the  exact  measurement  of 
the  distance  between  two  poles.  But  in  reality  magnetic 
poles  have  finite  dimensions  and  it  is  necessary  to  deter- 
mine the  mean  distance  between  them.  The  distance 
taken  is  that  between  two  points  so  situated  that  the  action 
between  the  two  poles  would  be  the  same  if  they  were 
concentrated  at  these  two  points.  We  therefore  think  of 
a  magnetic  pole  as  concentrated  at  a  point.  This  concep- 
tion is  no  more  strained  than  the  conception  of  centre  of 
gravitative  attraction  of  a  body,  where  we  consider  the 
whole  mass  of  the  body  as  concentrated  at  a  point. 

The  length  of  a  compound  pendulum  is  measured  in  a 
similar  way,  by  considering  that  the  mass  of  the  pendulum 
is  concentrated  at  such  a  point  that  the  time  of  oscillation 
is  not  changed. 


element  of  the  distance  ds,  divided  by  the  time  dt,  in  which  the  distance 
ds  is  traversed;  that  is,  velocity  equals  the  rate  of  change  of  length  with 

respect  to  time,  v  =  — -.     In  the  C.  G.  S.  system  velocity  is  measured  in 
ctt 

centimetres  per  second.     The  acceleration  of  the  body  is  the  rate  at  which 

the  velocity  is  changing;  that  is,  a  =  — -  =  — .     In  the  C.  G.  S.  system 

ctt       ctt 

acceleration  is  measured  in  centimetres  per  second. 

By  Newton's  first  law,  every  body  continues  in  a  state  of  rest,  or  of  uni- 
form motion  in  a  straight  line,  except  in  so  far  as  it  may  be  compelled  by 
impressed  forces  to  change  that  state.  Force  may  be  defined  as  that  which 
causes  or  tends  to  cause  a  change  in  the  velocity  of  a  body.  The  unit  of 
force  is  that  force  which  causes  a  unit  change  in  velocity  of  a  unit  mass 
in  unit  time,  that  is,  produces  unit  acceleration.  In  the  C.  G.  S.  system 
the  unit  of  force  is  the  dyne,  and  is  the  force  which,  when  acting  for  one 
second,  will  give  a  mass  of  one  gramme  a  velocity  of  one  centimetre  per 
second. 


20  INTRODUCTORY  TO   CIRCUITS 

LAW   OF  ATTRACTION. 

The  law  of  the  action  between  magnetic  poles,  as  ex- 
perimentally determined  by  Coulomb,  is  that  the  attraction 
or  repulsion  between  two  poles  is  inversely  as  the  square 
of  the  distance  between  them,  and  directly  as  the  product 
of  their  strengths  ;  that  is, 

m  mf 


where  m  and  m'  are  the  strengths  of  the  poles,  that  is,  the 
number  of  unit  poles  to  which  each  is  equivalent,  and  r 
the  distance  between  them. 

A  unit  pole  being  as  previously  defined,  the  sign  of 
variation  may  be  changed  for  one  of  equality  if  the  dis- 
tance r  is  measured  in  centimetres  and  the  force  F  is 
measured  in  dynes.  The  force  between  two  magnetic  poles 
is  then 

m  m' 

fi     —*_     , 


When  the  poles  considered  have  the  same  sign,  and  are 
both  north  poles  or  both  south  poles,  the  product  m  m'  is 
positive,  and  a  force  of  repulsion  has  the  positive  sign. 
Similarly,  a  force  of  attraction  has  a  negative  sign. 

INTENSITY   OF    A   FIELD   OF   FORCE. 

The  strength  of  a  magnetic  field  of  force  at  any  point  is 
measured  by  its  action  on  a  unit  positive  magnetic  pole  at 
that  point. 

If  we  could  place  a  free  magnetic  pole  in  a  magnetic 
field,  it  would  always  be  urged  in  a  certain  direction  ;  and,, 
if  free  to  move,  would  actually  move  in  this  direction. 
The  direction  in  which  a  positive  pole  would  be  urged  is 
called  the  positive  direction  of  the  line  of  force  which 


CONTAINING  RESISTANCE  AND  SELF  INDUCTION.      21 

passes  through  the  pole.  The  force  with  which  a  unit 
positive  pole  would  be  urged  at  any  point  of  a  magnetic 
field  is  the  strength  of  the  field  at  that  point,  and  is  usually 
denoted  by  H. 

Usually  it  is  found  that  H  varies  at  different  points  in 
the  field  ;  but  if  H  has  the  same  value  at  every  point,  both 
in  magnitude  and  in  direction,  the  field  is  said  to  be  uni- 
form. 

If  the  uniform  field  be  one  of  unit  intensity,  then  H  =  1, 
and  there  is  said  to  be  one  line  of  force  per  square  centi- 
metre ;  and  when  the  intensity  is  H,  there  are  H  lines  of 
force  per  square  centimetre.  Thus  the  intensity  of  a  mag- 
netic field  is  thought  of  as  being  determined  by  the  number 
of  lines  which  pass  through  one  square  centimetre  of  a 
surface  normal  to  the  direction  of  the  lines  of  force. 

As  an  example,  by  the  definition  of  a  unit  pole  the  in- 
tensity of  the  field  H  is  unity  at  a  distance  of  one  centime- 
tre from  the  pole.  If  a  sphere  be  described  about  the  unit 
pole  as  a  -centre,  having  a  radius  of  one  centimetre,  there  is 
consequently  one  line  of  force  passing  through  the  surface 
of  the  sphere  for  every  square  centimetre.  As  the  surface 
of  the  sphere  contains  4  TT  square  centimetres,  there  are  in 
all  4  TT  lines  of  force  that  emanate  from  a  unit  pole,  and 
4  7t  m  lines  from  a  pole  whose  strength  is  m. 

INDUCTION. 

The  number  of  lines  of  force  in  air  is  the  same  as  the 
number  of  lines  of  magnetizing  force.  In  a  magnetic  sub- 
stance, such  as  iron,  the  number  of  lines  of  force  is  greatly 
increased,  and  they  are  then  called  lines  of  magnetization, 
or  lines  of  induction. 

The  number  of  lines  of  induction  N,  passing  through 
any  area,  is  called  the  total  magnetic  induction  through 
this  area.  The  total  number  of  lines  of  force  per  squaro 


22  INTRODUCTORY  TO   CIRCUITS 

centimetre  of  area  normal  to  these  lines  is  called  the  in- 
duction per  square  centimetre,  or  simply  the  induction,  B. 
In  a  non-magnetic  medium  the  induction  B  is  equal  to  the 
magnetizing  force,  H.  In  a  magnetic  medium,  such  as  iron, 
the  magnetizing  force  produces  an  induction  B  greater 
than  H.  The  ratio  of  the  induction  to  the  magnetizing 

force  is  called  the  permeability,  // ;  that  is,  ji  =  -fi.. 

H 

A  current  of  electricity  flowing  in  a  circuit  always  pro- 
duces a  magnetic  field  in  the  surrounding  region.  The 
lines  of  force  which  constitute  this  field  are  always  closed 
lines  which  encircle  the  conductor.  The  total  number  of 
Hues  passing  through  the  area  bounded  by  a  closed  electric 
circuit  is  the  total  magnetic  induction  of  the  circuit.  As 
the  current  is  increased  in  strength,  the  intensity  of  the 
magnetic  field  at  every  point  is  increased,  and  if  there  is 
no  magnetic  substance  in  the  region,  the  intensity  of  the 
field  is  increased  in  direct  proportion  to  the  strength  of 
current. 

A  unit  current  is  defined  in  terms  of  the  intensity  of  the 
magnetic  field  which  it  generates.  A  unit  current  is  that 
which,  flowing  in  a  circuit  of  one  centimetre  radius,  acts 
on  a  unit  magnetic  pole,  placed  at  the  centre,  with  a  force 
of  one  dyne  per  centimetre  length  of  the  circumference. 
This  is  a  unit  of  current  in  the  C.  G.  S.  system.  Each  unit 
length  of  conductor  is  acted  upon  by  the  unit  pole,  placed 
at  the  centre,  with  a  force  of  one  dyne,  which  is  the  same 
force  as  that  by  which  a  unit  pole  would  be  acted  upon 
when  substituted  for  a  unit  length  of  the  conductor  at  the 
same  distance.  The  practical  unit  of  current,  the  ampere, 
is  one-tenth  of  the  C.  G.  S.  unit. 

NUMBER  OF  LINES  PROPORTIONAL  TO   CURRENT. 

We  have  seen  that  when  a  current  flows  through  a 
closed  circuit  a  field  is  set  up  consisting  of  a  definite  num- 


CONTAINING  RESISTANCE  AND  SELF  INDUCTION.      23 

ber  of  lines  threading  tlie  circuit.  If  there  is  no  magnetic 
substance  in  the  vicinity,  that  is  if  the  permeability  of  the 
surrounding  region  be  constant,  the  number  of  lines  pro- 
duced by  a  current  in  a  circuit  is  directly  proportional  to 
the  current,  and  any  change  in  the  current  produces  a  pro- 
portional change  in  the  number  of  lines  threading  the  cir- 
cuit. 

dN       di 
This  may  be  expressed  N  a  i,  and  —rr  <*  -?.     Since  N 

varies  as  i,  we  may  say 

N=Li, 

(1)     and  consequently  —^  —  L  —. 

The  coefficient  L  is  called  the  coefficient  of  self -induction, 
and  is  defined  by  the  equation  as  the  ratio  of  the  total  in- 
duction threading  a  circuit  to  the  current  producing  it. 
When  the  current  is  unity,  the  coefficient  of  self-induction 
is  equal  to  the  number  of  lines  produced  by  the  current. 
If  the  permeability  of  the  medium  surrounding  the  con- 
ductor is  constant,  this  will  be  the  value  of  L  for  all  values 
of  the  current,  and  L  will  be  constant.  Unless  a  high  degree 
of  magnetization  is  reached,  Z  is  approximately  a  constant 
for  a  given  circuit,  and  will  hereafter  be  so  considered. 

FARADAY'S  LAW  or  ELECTROMOTIVE  FORCE. 

When  a  conductor  is  moved  in  a  magnetic  field  so  as 
io  cut  lines  of  force  an  electromotive  force  is  produced  in 
the  conductor.  Faraday  showed  as  the  result  of  his  re- 
searches that  this  E.  M.  F.  produced  is  directly  propor- 
tional to  the  rate  of  cutting  the  lines  of  force,  and  is  in  a 
direction  at  right  angles  to  the  direction  of  motion  and  also 
to  the  direction  of  the  lines  of  force.  He  further  showed 
that,  if  the  magnetic  induction  through  any  closed  circuit 
be  varied  by  any  means,  an  E.  M.  F.  is  developed  in  the 


24  INTRODUCTORY  TO  CIRCUITS 

circuit  proportional  at  any  instant  to  the  rate  of  change 
(decrease)  of  the  magnetic  induction  at  that  instant.  In 
the  C.  G.  S.  system  of  units  this  experimental  lav,  may  be 
expressed  by  the  equation 

where  e  denotes  the  E.  M.  F.  developed  and  ^Tthe  magnetic 
induction  of  the  circuit.  This  means  that  a  C.  G.  S.  unit 
E.  M.  F.  is  developed  when  there  is  a  change  in  the  induc- 
tion of  the  circuit  at  the  rate  of  one  line  per  second.  The 
negative  sign  indicates  that  the  E.  M.  F.  is  induced  in  such 
a  direction  as  to  oppose  the  change  in  the  number  of  lines 
threading  the  circuit.  The  practical  unit  of  E.  M.  F.,  the 
volt,  is  10h  times  the  C.  G.  S.  unit  just  defined. 

OHM'S  LAW. 

An  electromotive  force  impressed  upon  a  closed  circuit 
causes  a  current  to  flow  which  depends  upon  the  resistance 
of  the  circuit.  Ohm  first  showed,  and  others  have  since 
verified  to  a  high  degree  of  accuracy,  that  with  a  constant 
E.  M.  F.,  the  current  is  directly  proportional  to  the  E.  M.  F. 
and  inversely  proportional  to  the  resistance  of  the  circuit.. 

Ohm's  law  may  be  expressed 


where  /  denotes  current  ;  E,  electromotive  force  ;  and  R, 
resistance.  Since  E.  M.  F.  and  current  are  already  inde- 
pendently defined,  the  unit  of  resistance  is  naturally  taken 
to  be  that  resistance  which  allows  a  unit  current  to  flow 
in  a  circuit  having  a  unit  impressed  E.  M.  F.  Ohm's  law 
may  then  be  expressed 


- 

R 


CONTAINING  RESISTANCE  AND  SELF  INDUCTION.      25 

From  tLe  relation  of  the  practical  units  of  E.  M.  F.  and 
current,  the  volt  and  the  ampere,  to  the  corresponding 
C.  G.  S.  units,  it  follows  that  the  practical  unit  of  resist- 
ance, the  ohm,  is  109  times  the  C.  G.  S.  unit. 

QUANTITY. 

A  unit  quantity  of  electricity  is  said  to  flow  in  a  circuit 
when  unit  current  flows  for  one  second.  When  a  current  / 
flows  in  a  circuit  for  t  seconds,  a  quantity  It  will  flow. 
And  in  a  short  interval  of  time  dt,  a  quantity  i  dt  will  flow, 
which  is  represented  by  dq,  thus  : 

dq 


q  representing  quantity. 

We  have  seen  that  for  a  constant  electromotive  force,  by 
Ohm's  law  the  current  equals  the  E.  M.  F.  divided  by  the 
resistance.  During  a  short  interval  of  time  dt,  any  E.  M.  F. 
may  be  considered  constant,  and  we  may  write 

ia 

i  =  ^p,  during  the  time  dt. 

The  capital  letters  E,  /,  and  Q  will  be  used  to  denote  a 
constant  electromotive  force,  current,  or  charge.  When 
these  are  variable  the  small  letters  e,  i,  and  q  will  be  used. 

When  a  closed  conductor  is  moved  from  one  position  to 
another  in  a  magnetic  field,  so  as  to  cause  the  number  of 
lines  of  force  included  by  the  circuit  to  change  from  one 
value  Nl  to  another  value  N^,  it  will  be  found  that  the 
quantity  of  electricity  which  flows  in  the  circuit  is  always 
a  definite  amount,  being  equal  to  the  change  in  the  number 
of  lines  j\72  —  Nlt  divided  by  the  resistance  of  the  circuit, 
and  is  entirely  independent  of  the  manner  of  the  change, 
and  of  the  time  occupied  in  making  the  change. 


20  INTRODUCTORY  TO   CIRCUITS 

This  will  be  evident  when  we  remember  Faraday's  law, 

dN 
e  =  —  -TT,  and  consider  that  the  only  E.  M.  F.  acting  in 

the  circuit  during  the   motion   of   the    conductor   is  this 

dN 
—  -j—  •     Hence  the  following  relations  are  true  : 

dq  _  .  _     e__        ± 
dt  ~~        ~  lR-    ~5 


(3)  Whence  Q  =     l~~R\ 

Here  Q  denotes  tire  sum  of  all  the  small  quantities,  or 
the  total  quantity  of  electricity  flowing  through  the  circuit 
during  the  motion  of  the  conductor,  and  is  seen  to  be  equal 
to  the  change  of  the  induction  through  the  circuit  divided 
by  the  resistance  of  the  circuit,  as  stated  above. 

The  earth  inductor  is  a  good  example  of  an  instrument 
which  depends  for  its  use  upon  the  principle  just  stated. 
When  a  ballistic  galvanometer  is  connected  with  an  earth 
inductor,  the  throw  of  the  galvanometer  is  proportional  to 
the  total  change  in  the  number  of  lines  of  force  included 
by  the  earth  inductor  coil  as  it  turns  from  one  position  to 
another,  provided  the  needle  does  not  start  to  swing  until 
the  whole  quantity  of  electricity  has  flowed?  through  the 
galvanometer. 

JOULE'S  LAW. 

The  fourth  and  last  great  experimental  law  to  be  men- 
tioned is  the  discovery  by  Joule  that  the  heat  liberated  by 
a  conductor  carrying  a  current  of  electricity  is  strictly  pro- 
portional to  the  product  of  the  square  of  the  current-strength 
and  the  resistance  of  the  conductor. 

Now  for  the  first  time  we  have  a  means  of  telling  how 
much  energy  is  required  to  send  a  current  of  any  desired 
.strength  through  a  conductor,  and  we  always  expect  to  find 


CONTAINING  RESISTANCE  AND  SELF  INDUCTION.       27 

some  source  for  the  supply  of  energy  when  we  see  a  current 
flowing  through  a  conductor. 

The  elementary  principles,  already  given,  upon  which 
the  system  of  electromagnetic  units  is  based,  are  deduced 
from  the  experimental  researches  of  Coulomb,  Faraday,  and 
Ohm.  When  a  current  flows  through  a  conductor  there  is 
always  heat  liberated  in  the  conductor  and  accordingly  a 
dissipation  of  energy.  It  therefore  requires  a  certain 
amount  of  energy  to  send  a  current  through  a  conductor. 
The  exact  amount  of  this  heating  effect  was  first  determined 
by  Joule.  The  results  of  his  experiments  show  that  the 
energy  liberated  per  second  in  the  form  of  heat  in  a  con- 
ductor carrying  a  current  of  electricity  is  strictly  propor- 
tional to  the  product  of  the  square  of  the  current-strength 
and  the  resistance  of  the  conductor.  Joule's  law  may  be 
expressed 


where  W  represents  the  energy  expended  per  second. 

The  energy  expended  in  the  time  t,  during  which  the 
current  is  constant,  is  P  R  t.  If  the  current  be  a  variable 
i,  it  may  be  considered  constant  for  the  time  dt,  and  so  in 
the  time  dt 

(4)  Energy  dissipated  in  heat  =  P  R  dt. 

When  all  the  energy  given  to  the  circuit  is  expended  in 
heat,  that  is  when  there  is  no  counter  E.  M.  F.  of  any  kind 
and  the  current  is  constant,  I  R  may  be  replaced  by  E, 
according  to  Ohm's  law,  and  the  energy  expended  per 
second  may  be  written 

Woe  EL 

This  becomes  more  definite  in  the  units  already  dis- 
cussed. If  a  conductor  carrying  a  current  /  be  placed  at 
right  angles  to  the  lines  of  force  in  a  uniform  field  of 


28  INTRODUCTORY  TO  CIRCUITS 

intensity  H,  each  unit  of  length  will  be  acted  upon  with  a 
force  H  /.  If  I  be  the  length  of  the  conductor,  the  force 
will  be  I  H  /.  When  moved  with  a  velocity  v  against  this 
force,  work  will  be  performed  at  the  rate  of  I  H  Iv  ergs  per 
second,  or 

W=l\\Iv. 

This  must  be  equal  to  the  rate  at  which  work  is  done  in 
generating  a  current  /,  by  moving  the  conductor  through 
the  field.  The  conductor,  when  moving  with  a  velocity  v, 
cuts  I  H  v  lines  per  second,  and  so  produces  an  E.  M.  F. 

E=l\\v. 

Substituting  above,  we  see  that  the  rate  at  which  energy 
is  expended  in  a  circuit  at  any  time  is  equal  to  the  product 
of  the  current  and  the  electromotive  force  at  that  time, 

W=EL 

This  is  seen  to  be  equivalent  to  Joule's  law  above  and  is 
equally  true  for  C.  G.  S.  and  for  practical  units.  In  the 
C.  G.  S.  system,  energy  is  measured  in  ergs  and  the  equa- 
tion expresses  the  fact  that  energy  in  ergs  is  equal  to  the 
product  of  current  and  E.  M.  F.  in  C.  G.  S.  units.  The 
practical  unit  of  energy  is  the  joule  and  is  so  denned  that 
the  equation  W  =  E  f,  true  in  ergs  and  other  C.  G.  S.  units, 
shall  be  also  true  in  practical  units — the  joule,  the  volt,  and 
the  ampere.  The  equation  is  then  interpreted  as  meaning 
that  energy  in  joules  is  equal  to  the  product  of  current  and 
E.  M.  F.  in  amperes  and  volts.  From  the  relation  already 
given  between  the  ampere  and  volt  and  their  corresponding 
C.  G.  S.  units,  the  joule  equals  10'  times  the  C.  G.  S.  unit 
the  erg. 

The   rate   of   ivork  is  in  electrical  terms  expressed  in 
watts  :  one  watt  equals  one  joule  per  second.     The  common 


CONTAINING  RESISTANCE  AND  SELF  INDUCTION.      29 

English  unit  of  rate  of  work  is  the  horse-power  :  one  horse- 
power equals  745.9  watts. 

The  rate  at  which  energy  is  imparted  to  a  circuit  multi- 
plied by  the  time  is  the  total  energy  imparted  during  that 
time.  If  there  is  a  variable  E.  M.  F.,  e,  from  any  source 
whatever  given  to  a  circuit,  and  a  current  i  flows,  the  energy 
imparted  to  the  circuit  in  the  time  dt  from  the  source  of 
this  E.  M.  F.  is  the  product  e  i  dt.  Thus : 

(5)  Energy  imparted  to  a  circuit  =  ei dt. 

This  enables  us  to  ascertain  the  energy  possessed  by  a 
magnetic  field.  By  Faraday's  law,  when  the  magnetic 
induction  through  any  closed  circuit  changes  from  any 
cause  whatsoever,  there  is  always  an  electromotive  force 
given  to  the  circuit  which  is  equal  to 

dN  _          di 

~~dt~-    ~Ldt 

This  E.  M.  F.  is  due  to  the  existence  of  the  magnetic 
field.  In  creating  the  field,  an  equal  and  opposite  E.  M.  F., 

di 

L  77 ,  is  necessary  to  drive  the  current.     The  work  which 
dt 

this  force  does  is  equal  to  the  product  of  the  force,  the 
current  which  flows  in  the  circuit,  and  the  time  dt ;  as  ex- 
plained above.  The  change  in  the  energy  possessed  by  a 
magnetic  field  in  the  time  dt  is,  therefore, 

.dN.         ,  .di.. 

i-=-  dt  =  Li^-dt. 

at  at 

(6)  Energy  expended  in  the  magnetic  field  =  L  i-*-  dt. 

CAJV 

The  change  in  the  induction  through  any  circuit  may 
be  due  to  any  external  cause,  as  moving  magnets,  or  it  may 
be  due  to  a  change  in  the  current  itself.  When  the  change 
is  due  to  a  change  in  the  current,  an  increase  in  the 


30  INTRODUCTORY  TO   CIRCUITS 

v 

strength   of    the    current   increases    the    energy    of    the 

magnetic  field ;  and  positive  work  is  done  by  the  current 

in  creating  the  field.     When   the  current   decreases,  the 

energy  of  the  field  decreases,  and  negative  work  is  done  by 

-  the  current  on  the  field ;  for,  when  the  current  decreases 

with  the  time,  -^-  is  negative.     To  say  that  the  current  is 

r*  ttt 

doing  negative  work  is  equivalent  to  saying  that  the 
magnetic  field  in  decreasing  is  imparting  energy  to  the 
circuit.  Thus  we  see  that  the  energy  may  be  stored  up  in 
a  magnetic  field,  and  that  this  is  not  dissipated,  but  is 
returned  to  the  circuit  when  the  field  is  diminished  in 
strength. 

To  find  the  expression  for  the  total  energy  of  a  mag- 
netic field  which  is  due  to  a  current  i  flowing  in  a  circuit, 
we  need  merely  find  the  sum  of  all  the  small  quantities  of 
energy  imparted  to  the  field  as  the  current  is  increased 
from  zero  to  its  final  value  / :  this  is  found  to  be 


(7) 


THE  EQUATION   OF  ENERGY. 


If  e  represents  the  impressed  E.  M.  F.  given  to  a  circuit 
which  has  a  resistance  R  and  a  coefficient  of  self-induction 
L,  we  have  seen  [equation  (5)]  that  the  total  energy  given 
to  the  circuit  from  the  source  is  e  i  dt. 

A.  part  of  this  energy  supplied  is  dissipated  in  heating 
the  conductor,  and  in  the  time  dt  is  equal  to  Ri*dt  [equa- 
tion (4)].  A  second  part  is  stored  up  in  the  magnetic  field, 

and  in  the  time  dt  is  equal  to  L  i  -r  dt  [equation  (6)].    These 

two  ways  ar-e  the  only  ones  in  which  the  energy  of  the 
source  is  used,  under  the  hypothesis  made  that  the  circuit 
contains  no  statical  capacity  or  counter  electromotive  force 


CONTAINING  RESISTANCE  AND  SELF  INDUCTION.      31 

of  any  kind  other  than  that  due  to  the  field,  but  only 
contains  a  resistance  R  and  a  self-induction  L. 

By  the  principle  of  the  conservation  of  energy  we  may, 
therefore,  say  that  the  energy  supplied  to  the  circuit  is 
the  sum  of  the  energy  dissipated  in  heat  and  the  energy 
expended  on  the  field. 

We  have,  therefore,  the  equation  of  energy : 

(8)  eidt  =  Biidt  +  Li~dt. 

When  each  member  of  the  equation  of  energy  is  divided 
by  i  dt,  we  obtain 

This  is  an  equation  of  electromotive  forces :  e  is  the  E.  M.  F. 
of  the  source  impressed  upon  the  circuit,  R  i  the  E.  M.  F. 

necessary  to  overcome  the  ohmic  resistance,  and  L  -r,  the 

Cut 

E.  M.  F.  equal  to  the  E.  M.  F.  of  self-induction. 


CHAPTER  II. 

ON  HARMONIC  FUNCTIONS. 

CONTENTS  :— Harmonic  E.  M.  F.  often  assumed.  Simple  harmonic  motion. 
Amplitude.  Period.  Angular  velocity.  Frequency.  Epoch.  Phase. 
Lag.  Advance.  Graphical  representation  of  simple  harmonic  functions. 
Average  value  of  ordinates  of  sine-curve.  Value  of  mean  square  of 
ordinates  of  sine-curve.  Periodic  functions  composed  of  several  simple 
sine-functions  of  like  period, — of  unlike  period.  Fourier's  theorem. 

IF  a  conductor  revolves  with  uniform  velocity  about 
some  fixed  axis  in  a  uniform  field,  the  rate  at  which  it  cuts 
the  lines  of  force  is  different  at  different  parts  of  the  revo- 
lution and  varies  directly  as  the  sine  of  the  angle  of  rota- 
tion. The  electromotive  force  set  up  in  the  conductor  at 
any  instant  is  numerically  equal  to  the  rate  of  cutting  lines 
at  that  instant  and  is  accordingly  a  sine-function  of  the 
angle  of  rotation  and,  since  the  rotation  is  uniform,  a  sine- 
function  of  the  time.  Inasmuch  as  the  assumption  of 
such  an  electromotive  force  often  closely  approximates  to 
the  truth,  and  since,  as  will  be  shown  later,  any  electro- 
motive force  whatever  may  be  expressed  as  a  sum  of  terms 
each  of  which  is  a  sine-function  of  the  time,  it  has  been 
found  convenient  to  express  electromotive  forces  in  terms 
of  sines. 

In  order  that  sine-functions  may  be  clearly  understood 
when  used  in  the  following  chapters,  it  is  considered 
advisable  to  digress  and  devote  the  present  chapter  to  the 

discussion  of  harmonic  or  sine-functions. 

32 


ON  HARMONIC  FUNCTIONS. 


HARMONIC  MOTION. 


33 


If  a  point  moves  uniformly  around  the  circumference  of 
a  circle,  the  motion  of  the  projection  of  that  point  upon 
any  fixed  diameter  is  said  to  be  harmonic.  The  radius  of 
the  circle  is  called  the  amplitude  of  the  motion,  and  is 
designated  by  a.  The  time  T  of  making  one  complete 
revolution  is  called  the  period.  Positive  rotation  will  be 
€onsidered  as  counter-clockwise. 

If  a  uniformly  revolving  radius  of  a  circle  is  projected 
upon  any  fixed  diameter,  its  projection  is  said  to  vary 
harmonically.  The  maximum  value  of  this  projection  is 
the  amplitude,  or  radius  of  the  circle.  This  is  represented 
in  Fig.  1.  P  is  a  point  moving  uniformly  about  the  centre 


FIG.  1. — HAKMONIC  MOTION. 


0,  and  OP'  is  the  projection  of  the  radius  OP  upon  the 
fixed  diameter  BD.  When  OP  is  in  the  position  OA  at 
right  angles  to  BD,  the  projection  OP'  is  zero  ;  and  when 
OP  is  in  the  position  OB,  the  projection  OP'  has  its 
maximum  value  and  is  equal  to  the  radius  OP.  The 
projection  is  again  zero  at  0(7,  and  a  negative  maximum 
at  OD. 


34  ON  HARMONIC  FUNCTIONS. 

The  angular  velocity  of  the  point  P  is  denoted  by  GOT 
and  at  any  point  is  GO  =.  -=-.  Since  the  motion  of  the  point 

is  uniform,  GO  —  — ,  or  GO  t  =  0,  where  0  is  the  angle  described 

v 

in  the  time  t     As  the  time  occupied  in  describing  a  circum- 

2?r  2?r 

ference  is  T,  the  uniform  velocity  GO  =  -^ ;  hence  0  =  -^-£. 

The  second  is  taken  as  the  unit  of  time.  The  number  of 
revolutions  made  by  the  moving  point  P,  in  one  second  is 

-=,  and  is  called  the  periodicity  or  frequency,  often  denoted 
by  n.  The  frequency  is  the  reciprocal  of  the  period,  i.e., 
n  =  -ip:  It  is  evident  that  the  angular  velocity  may  be 

expressed  in  terms  of  the  frequency;  thus,  GO  —  %7rn,  and 
therefore,  0  =  %nnt. 

If  we  begin  to  count  the  time  from  the  point  A  (Fig.  1), 
where  the  projection  of  OP'  is  zero,  denoting  OP'  by  y,  we 
have  at  any  time 

y  =  a  sin  0  =  a  sin  GO  t, 

where  a  denotes  the  amplitude  and  0  the  angle  described 
in  the  time  t ;  y  is  an  harmonic  or  sine-function  of  the  angle 
0  or  the  time  t. 

Suppose  that  the  time  is  counted  from  some  point  Q, 
Fig.  2,  other  than  the  point  A  at  which  the  projection  of  OP 
is  zero.  There  is  an  angle  6,  called  the  angle  of  epoch,  be- 
tween the  point  from  which  time  is  reckoned  and  the  point 
at  which  the  projection  of  the  radius  is  zero.  The  time  in 
which  this  angle  is  described  is  called  simply  the  epoch. 
As  before,  the  angle  0  is  that  described  in  the  time  t.  The 
angle  (0  -f-  0),  through  which  the  point  P  has  revolved 
from  the  point  A  where  the  projection  of  OP  is  zero,  is 
called  the  angle  of  phase  or  briefly  the  phase.  More  strictly 


ON  HARMONIC  FUNCTIONS. 


35 


defined,  the  phase  is  the  ratio  of  the  arc  PA  to  the  circum- 
ference of  the  circle. 

If  we  denote  by  y  the  projection  of  Of  upon  BD,  and 
count  time  from  Q, 

(10)  y  =  a  sin  (0  +  0)  =  a  sin  (cot  -f  #). 

When  6  is  positive, — that  is  when  it  is  in  the  positive  or 
counter-clockwise  direction  from  A,  as  in  Fig.  2, — it  is  often 
called  the  angle  of  advance.  When  0  is  negative, — and  the 


FIG.  2.— SIMPLE  SINE-CURVE. 


time  is  counted  from  some  point  Qf  at  a  distance  0  in  the 
negative  or  clockwise  direction  from  A, — it  is  often  called 
the  angle  of  lag.  When  the  angle  of  phase  is  zero,  OP  co- 
incides with  OA  and  the  projection  y  =  0.  When  the 
phase  is  90°,  the  projection  is  a  maximum,  and  y  =  -j-  a. 
At  180°,  again,  y  =  0  ;  and  at  270°,  y  =  —  a,  a  maximum  in 
the  negative  direction.  This  cycle  is  repeated  every  revo- 
lution. 

In  the  equation  y  =  a  sin  (GO  t  -f-  #),  the  amplitude,  a, 
angular  velocity,  GO,  and  angle  of  epoch,  0,  are  constants, 
and  the  variable  y  is  said  to  be  expressed  as  a  simple  sine- 
function  of  the  variable  t.  The  time  t  is  directly  propor- 
tional to  the  angular  distance  passed  through.  A  variable 
whose  value  at  any  time  can  be  expressed  as  a  constant 
multiplied  by  the  sine  of  an  angle  changing  uniformly  with 


36  ON  HARMONIC  FUNCTIONS. 

the  time,  is  called  a  simple  sine-function,  or  simple  har- 
monic function  of  the  time. 

In  Fig.  2,  y  is  plotted  as  a  sine-function  of  t.  At  any 
time,  £,  when  the  revolving  point  has  the  position  P,  y  has 
a  value  OP7. 

Angle  of  epoch  —  AOQ  —  6. 

Time  of  epoch   =  a'q'. 

Angle  described  in  time  t  =  QOP  —  <p  =  o$t. 

Angle  of  phase  =  0  -f-  (angle  of  epoch)  =  0  -f-  0. 

Time  of  phase  =  t  -\-  (time  of  epoch)  =  t  -\-  a'q'. 

Amplitude          =  OA '.  =  OB  =  a. 

When  the  term  "  harmonic  function"  or  "  sine-function"' 
or  "  sine-curve"  is  used,  such  a  function  or  curve  as  shown 
in  Fig.  2  is  meant. 

TO  FIND  THE  AVEKAGE  VALUE   OF   THE   ORDINATE   OF   A 

SINE  CUE VE. 

A  sine-curve  repeats  itself  symmetrically  and  the  aver- 
age ordinate  for  the  whole  period  is,  therefore,  algebraically 
zero,  as  it  is  negative  and  positive  alternately  for  equal 
intervals  of  time.  We  can,  however,  obtain  the  average 
value  for  one  half  a  period  and,  since  the  second  half  is  a 
repetition  of  the  first  half  with  sign  reversed,  this  will  give 
the  arithmetical  average  value  for  the  whole  period. 

The  average  ordinate  is  equal  to  the  sum  of  all  the 
vertical  elements  of  area  divided  by  their  number,  or,  what 
is  the  same  thing,  it  is  equal  to  the  area  included  between 
the  curve  and  the  axis  of  abscissae,  divided  by  the  base. 
For  half  a  period  the  limits  are  0  and  n,  so  the 

/   ydq       1   p* 
Average  Or  din  ate  —  ~% =  -  /  y  dx. 


• 

ON  HARMONIC  FUNCTIONS.  •     87 

But  for  a  sine- curve,  y  —  a  sin  x  ;  therefore 

2a 


Average  Ordinate  =  —  /    sin  xdx  =  —     — 

TTt/0  TT^ 


cos  cc  =  — 
n 


o 
But   a  is  the  maximum  ordinate,  and  —  =  .6369;  so  we 

7T 

may  write 

Average  Ordinate 

ll-L)  =jv  -  :  --  -p^:  —  :p  -  ;  —   —   .OOUt/, 

Maximum  Ordinate 
which  determines  the  value  of  the  average  ordinate. 

TO   FIND   THE   VALUE   OF   THE    MEAN    SQUARE   OF   THE   OKDINATES 
OF  A  SINE-CUEVE. 

Although  it  is  often  useful  to  know  the  value  of  the 
average  ordinate  of  a  sine-curve,  it  is  more  often  desirable 
to  know  the  value  of  the  mean  square  of  the  ordinates,  or 
the  square  root  of  the  mean  square.  Since  the  square  of 
an  ordinate  is  positive  irrespective  of  the  sign  of  the 
ordinate,  we  can  find  the  mean  square  of  the  ordinates  by 
integrating  for  the  whole  and  not  for  half  a  period  as  was 
necessary  in  finding  the  average  ordinate. 

/>» 
y2  dx     a«     _*„. 
jyjeau  square  01  y  —  -—73=—  =rsr-  /     sin2  xdx. 

~dx       ^ 

<A 

.  2          1      cos  2x      mi        . 
But  sin8  x  =  =  ---  -  --     Therefore 


*  .  "  x      sin  2x 

sin8  xdx  =  \      o  --  -A  —  =  it. 

Lo  2          4 


Substituting  n  for  the  integral  above,  we  have 
(12)  Mean  Square  of  y  =  ^-  x  TT  =  ^. 


38  ON  HARMONIC  FUNCTIONS. 

The  square  root  of  the  mean  square  of  the  ordinates  is, 
therefore, 


This  means  that  the  square  root  of  the  mean  square  of 
the  instantaneous  values  of  y,  which  varies  harmonically 
with  the  time,  is  equal  to  .707  of  the  maximum  ordinate. 

The  square  root  of  the  mean  square  of  the  instantaneous 
values  of  a  variable  current  or  electromotive  force  is  called 
the  virtual  current  or  electromotive  force  and  is  equal  to  .  707 
times  the  maximum  value  of  the  current  or  electromotive 
force. 

Inasmuch  as  the  heating  and  dynamometer  effects  of 
any  current  depend  directly  upon  its  mean  square  value, 
this  virtual  value  is  of  much  more  importance  than  the 
average  value  in  the  measurement  of  an  alternating  current. 

PERIODIC     FUNCTIONS      COMPOSED      OF      SEVERAL      SIMPLE     SINE- 

FUNCTIONS. 

A  single-valued  function  is  one  which  has  but  one  value 
at  any  one  point  of  time,  as  represented  in  Fig.  3.  A  mul- 

tiple-valued function  is  one  which 
may  have  more  than  one  value  at 
one  point  of  time,  as  represented 
in  Fig.  4.  A  periodic  function  is 
one  which  repeats  itself  after  a 


FIG.  3.— SINGLE-VALUED 
FUNCTION. 


definite   time   or   period.     If   any 
number  of  simple  sine-functions  of 
the  same  period  be  added,  the  resultant  sum  will  be  a 


X-Axis 

FIG.  4. — MULTIPLE- VALUED  FUNCTION. 
simple  sine-function  of  the  same  period.    This  is  rigorously 


ON  HARMONIC  FUNCTIONS. 


shown  in  Chap.  XIV.,  Part  II. ,  for  the  addition  of  two 
simple  sine-functions,  as  illustrated  in  Fig.  47 ;  and  it  is  evi- 
dent that,  if  true  for  the  addition  of  two,  it  is  true  for  the  ad- 


FIG.  5.— ADDITION  OF  SIMPLE  HARMONIC  CURVES  OF  SAME  PERIOD. 
dition  of  any  number  of  simple  sine -functions.    An  example 
of  the  addition  of  two  simple  sine-functions  of  the  same 


FIG.  6.— ADDITION  OF  SIMPLE  HARMONIC  CURVES  OF  DIFFERENT  PERIODS. 


40  ON  HARMONIC  FUNCTIONS. 

period  is  shown  in  Fig.  5.     The  resultant  curve,  represented 
by  the  heavy  line,  is  likewise  a  sine-curve. 

If  a  number  of  simple  sine-functions  of  periods  which 
are  different  but  commensurable,  are  added  together,  the 
resultant  sum  is  a  function  which  is  periodic,  but  not  har- 
monic, with  a  period  equal  to  the  least  common-multiple  of 
the  periods  of  the  several  component  sine-functions.  The 
two  heavy  curves  in  Fig.  6  are  obtained  by  adding  two 
simple  sine-curves  of  the  same  amplitude  and  with  periods 
in  the  ratio  1  :  2.  The  equation  for  the  lower  heavy  curve  i& 

y  =  a  sin  GO  t  -f-  a  sin  2  GO  t, 

the  two  component  curves,  shown  by  dotted  lines,  being 
zero  at  the  start.     The  upper  curve  has  the  equation 


y  =  a  sin  GO  t  -f-  a  sin  ^2  a?  t  -f-  ?J  , 

the  component  dotted  curves  never  being  zero  at  the  same 
time. 

The  addition  of  two  sine-curves  with  different  amplitudes 


FIG  7.— ADDITION  OP  SIMPLE  HARMONIC  CURVES  OP  DIFFERENT  PERIODS. 

and  with  periods  in  the  ratio  1  :  3  is  illustrated  in  Figs.  7 
and  8.     The  component  curves  in   Fig.  7  have  no  phase 


ON  HARMONIC  FUNCTIONS.  41 

difference  at  the  start  and  the  resultant  curve  represents 
the  equation 

y  =  a  sin  GO  t  -/-  6  sin  3  GO  t. 

The  curve  in  Fig.  8  represents  the  equation 
y  —  a  sin  GO  t  -f  b  sin  (3  GO  t  —  6). 

By  adding  a  number  of  component  simple'  sine-curves 
with  different  periods  and  amplitudes,  resultant  periodic 


FIG.  8. — ADDITION  OF  SIMPLE  HARMONIC  CURVES  OF  DIFFERENT  PKIUOOS 

curves  of  all  manner  of  forms  are  obtained.  Fourier  has 
shown  that  any  single-valued  periodic  curve  may  be  built 
up  by  combining  a  number  of  simple  sine-curves.  Analyt- 
ically this  means  that  any  single-valued  periodic  function 
may  be  expressed  as  the  sum  of  a  series  of  sine-terms ;  thus, 

y  =f(x)  —  A  sin  a  x  -f-  B  sin  ft  x  -f-  C  sin  y  x  -{-  .  .  .  etc., 

where  /  is  a  single-valued  function.  This  is  true  for  any 
single-valued  periodic  function,  even  one  represented  by 
an  irregular  series  of  straight  lines. 


CHAPTER  III. 

CIRCUITS  CONTAINING  RESISTANCE  AND  SELF  INDUCTION. 

CONTENTS:— Equations  of  energy  and  E.  M.  F.'s.  Criterion  of  integra- 
bility.  General  solution  when  e  =  f(t). 

Case  I.  E.  M.  F.  suddenly  Removed.  Solution  from  differential  equa- 
tion,— from  general  solution.  Geometric  construction  of 
logarithmic  curve. 

Case  II.  E.  M.  F.  suddenly  Introduced.  Solution  from  differential 
equation, — from  general  solution. 

Case  III.  Simple  Harmonic  E.  M.  F.  Solution  from  general  equation. 
Impedance.  Lag.  Effect  of  exponential  term  at  "  make." 

Case  IV.  Any  Periodic  E.  M.  F.  Sum  of  two  sine-functions.  Sum  of 
any  number  of  sine-functions. 

IN  the  first  chapter  the  equation  of  energy  for  a  circuit 
containing  self-induction  and  resistance  was  derived,  and 
from  it  the  equation  of  electromotive  forces 

di 
(9)  e=Ri+Ljt; 

that  is,  the  electromotive  force  applied  to  the  circuit  is 
equal  to  the  sum  of  the  electromotive  force  necessary  to 
overcome  resistance  and  the  electromotive  force  necessary 
to  overcome  the  counter  electromotive  force  of  self-induc- 
tion. 

This  equation  of  electromotive  forces,  being  regarded 
as  a  differential  equation  containing  three  variables  e,  i, 
and  I  (of  which  the  general  type  of  the  first  order  is 

(13)  Pdx+Qdy  +  Sdz  =  0, 

42 


RESISTANCE  AND  SELF  INDUCTION.  43 

where  P,  Q,  and  S  are  any  functions  of  x,  y,  and  z)  does 
not  satisfy  the  condition  of  integrability.  That  condition, 
which  must  hold  true  when  there  exists  a  single  integral 
equation  of  which  (13),  or  a  multiple  of  (13),  is  the  exact 
differential,*  is 

dQ      dS\          IdS     dP\  dP     dQ 


If  we  put  (9)  in  the  form  of  (13),  we  have 
Ode  —  Ldi  +  (e  —  Ri)  dt  =  0. 

Here  e,  i,  and  t  correspond  to  x,  y,  and  z  respectively, 
and  P  =  0,  Q  =  —  L,  S  =  e  —  Ri. 

The  criterion  of  integrability  reduces  to 

-  L  (1  -  0)  =  0,     or     —  L  =  0, 

and  is  not  satisfied. 

The  meaning  of  this  is  that,  unless  some  relation  exists 
between  two  or  more  of  the  variables,  there  is  no  single 
equation  of  which  (9)  is  the  exact  differential. 

We  know  that  the  impressed  E.  M.  R,  e,  has  one  single 
value  at  any  particular  point  of  time,  and  may  therefore  be 
expressed  as  a  function  of  the  time  thus, 

(14)  e  =/(«), 

where  /is  any  arbitrary  single-valued  function. 

By  equating  (14)  to  (9)  the  equation  (9)  of  E.  M.  F.'s  is 
reduced  to  a  linear  equation,  having  constant  coefficients 


with  the  second  member  equal  to  —j*     Thus, 


*  See  Johnson's  Differential  Equations,  p.  270. 


44  CIRCUITS  CONTAINING 

The  general  type  of  this  equation  is 

(16)          ..V,. 


where  P  and  Q  may  be  functions  of  x  only.  The  solution 
of  equation  (16),  which  is  a  linear  differential  equation  of 
the  first  order,*  is 

y  =  e  -fp**j*efpdx  Qdx  +  ce  ~fpdx. 

e  denotes  the  base  of  the  Naperian  system  of  logarithms 
and  is  equal  to  2.718.  c  is  the  arbitrary  constant  of  in- 
tegration. Both  of  these  letters  will  be  thus  used  when- 
ever they  occur. 

With  the  particular  values  of  the  coefficients  in  (15)  its 
solution  is,  therefore, 


Rt  Rt  Rt 

L 


(17)  i  =  ~i  rfeL  f(t)  dt  +  ce 

This  is  the  general  solution  for  the  current  flowing  in  a 
circuit  containing  resistance  and  self-induction  and  any 
impressed  E.  M.  F. 

The  integration  indicated  in  (17)  can  only  be  performed 
when  we  assume  e  to  be  some  particular  function  of  t. 
We  proceed  then  to  assume  several  ways  in  which  the 
E.  M.  F.  varies  with  the  time. 

CASE  I.    DYING  AWAY  OF  CURRENT  ON  EEMOVAL  or  E.  M.  F. 

FROM  A   CIRCUIT  CONTAINING  RESISTANCE  AND  SELF-INDUC- 
TION. 

Suppose  that  a  current  has  been  flowing  in  a  circuit 
until  it  has  reached  its  steady  state,  and  that  the  source  of 
E.  M.  F.  is  then  suddenly  removed  while  the  resistance 

*  See  Johnson's  Differential  Equations,  p.  31. 


RESISTANCE  AND  SELF  INDUCTION.  45 

and   self-induction   remain    the   same.     The    equation   of 
electromotive  forces  (9)  becomes,  under  this  hypothesis, 


The  solutioc  of  this  equation  is  readily  found  since  the 
variables  admit  of  separation.     Thus, 

di  R 

i          Rt 

•'•  lo&r  =  -T' 

o  _/_> 

or 

Rt 


=  c  e 


L 


The  constant  of  integration  c  is  determined  by  the  par- 
ticular supposition  introduced  that  when  we  begin  to  count 
the  time,  the  current  has  its  steady  value  /.  This  gives 
c  —  L  Hence  we  have 

_Rt 

(18)  i  =  Ie~L. 

Referring  to  the  general  solution  (17),  we  might  have 
written  (18)  at  once.  For  as/(f)  =  0,  [see  (14),]  the  integral 
vanishes,  and  we  have  (18)  as  an  immediate  result. 

This  equation  (18)  is  graphically  represented  in  Fig.  9, 
where  the  ordinates  represent  the  values  of  the  current  at 
any  time  after  the  E.  M.  F.  is  removed.  The  self-induction 
of  the  circuit  prevents  the  current  from  falling  immediately 
to  zero.  It  is  evident  that  it  would  do  so  if  there  were  no 
self-induction  from  equation  (18) ;  for,  if  we  make  L  =  0, 
i  becomes  zero.  The  current  which  flows  after  the  removal 
of  the  E.  M.  F.  is  called  the  extra  current  of  self-induction. 
The  energy  required  to  cause  such  a  current  to  flow  is  that 
energy  which  was  previously  stored  up  in  the  field  and  is 


46 


CIRCUITS  CONTAINING 


now  returned  to  the  circuit.     When  t  has  the  value  -=,  the 

H 

exponent  of  e  becomes  minus  unity,  and  we  have  the  rela- 


Seconds 


YIQ.  9. — CURVE  SHOWING  THE  DYING  AWAY  OF  CURRENT  AT  ANY  TIME 
AFTER  THE  REMOVAL  OF  THE  IMPRESSED  E.  M.  F.  FROM  A  CIRCUIT 
WHOSE  RESISTANCE  R  is  .1  OHM  AND  COEFFICIENT  OF  SELF-INDUC- 
TION L  is  .01  HENRY. 

tion  T  =  e  =  2.71828.     -«  represents,  therefore,  the  time 


that  it  takes  for  the  current  to  fall  to  one  eth  part,  that  is  to 
o  r 

a. 


*  its  original  value.  This  is  sometimes  called  the 
time-constant  of  the  circuit,  and  denoted  by  T,  that  is 
-n  =  T.  The  curve  represents  an  exponential  function  of 

the  time  and  approaches  the  x-axis  as  an  asymptote.  This 
means  that  the  current  becomes  smaller  and  smaller,  but 
is  never  zero  until  an  infinite  time  has  elapsed. 

GEOMETRICAL     METHOD     OF     CONSTRUCTING     THE      LOGARITHMIC 

CURVE. 

The  following  method  shown  in  Figs.  10  and  11  will  be 
found  to  be  a  convenient  way  to  construct  a  curve  graphi- 
cally whose  equation  is  of  the  form 

(19)  y  =  ce~"*, 

where  c  and  a  have  any  real  values  whatever. 


RESISTANCE  AND  SELF  INDUCTION. 


47 


Lay  off  OA  equal  to  c.     Then  OA  is  the  value  of  y 
when  x  =  0  and  may  be  called 
y0 ;  that  is,  y0  =  c  —  OA. 

Lay  off  OB  equal  to  c  e""*1- 
Then    OB  is   the  value  of  y    o 
when    #  =  xlt    and    may    be 
called  yl ;  that  is,  yl  —  ce""*1 
=  03. 


_ 

TT  2/°         OA  aXl 

nence  —  —  -^  -•  —  —  e 


FIG.  10.  —  GRAPHICAL  METHOD  OP 
CONSTRUCTING  A  LOGARITHMIC 


2/i 


OB 


If  arcs  AA  and  .B.Z?'  are  described  from  the  centre  0,  and 
a  line  BG  drawn  parallel  with  A'B',  thence  another  line 
CD  drawn  parallel  with  AB,  and  so  on,  lines  parallel 
with  A'B'  and  with  AB  being  alternately  drawn,  as  in  the 
figure,  then  the  distances  ~OA,  OB,  OC,  OD,  etc.,  will  rep- 
resent the  values  of  y  respectively  as  x  takes  the  values 
0,  xl9  2a?lf  Bxl9  etc.  For  if  y0,  ylt  ya,  y^,  etc.,  denote  the 
values  of  y  when  x  takes  the  values  0,  xl9  %xl9  Sxlt  re- 
spectively, we  have 


2/o  = 


2/3 


Hence 


-  =       =  etc. 

2/2     2/3    2/4 


48 


CIRCUITS  CONTAINING 


From  the  construction  of  the  figure,  and  remembering 
that  OA  =  y0  and  OB  =  y, ,  we  see 


_  OB 
~OB  ^~OC 


00       OD 


Hence 


FIG.  11.— LOGARITHMIC  CURVE. 

Therefore  to  construct  the  curve  y  =  ce~ax.  Fig.  11, 
we  may  proceed  as  follows :  Upon  two  intersecting  lines, 
as  in  Fig.  10,  lay  off  the  distances  y0  =  c,  and  y1  =  c  e  ~  aXl, 
which  latter  must  be  calculated,  and  obtain  the  values  of 
00,  OD,  etc.,  as  described.  Then  y0,  yl9  y9,  etc.,  or  OA, 
OB,  00,  etc.,  will  be  the  successive  ordinates  of  the  loga- 
rithmic curve,  Fig.  11,  at  distances  Q,.xlt  %xl9  3xlt  etc.,  and 
the  curve  may  be  drawn. 

CASE  II.  ESTABLISHMENT  OF  A  CURRENT  ON  INTRODUCTION 
OF  A  CONSTANT  ELECTROMOTIVE  FORCE  INTO  A  CIRCUIT 
CONTAINING  EESISTANCE  AND  SELF-INDUCTION. 

Suppose  a  source  of  constant  E.  M.  F.  is  suddenly 
introduced  into  a  circuit  of  resistance  R  and  self-induction 
L.  The  differential  equation  in  this  case  is 


(20) 


RESISTANCE  AND  SELF  INDUCTION.  49 

where  E  is  a  constant.     The  variables  may  be  separated 
here  as  in  the  previous  case,  thus  : 

di  R  , 

~E  ~  ~  X     ' 


I/. 
and 


- 

Therefore  i  =  -75  —  (-  c  e   L 

ja 


The  constant  of  integration,  c,  is  determined   by  the 
dition  that,  when  t  = 
We  have  then  as  a  result, 


TjJ 

condition  that,  when  t  =  0,  i  =  0,  and  therefore  c  =  —  -=> 


-e^ 


Keferring  to  the  general  solution  (17)  we  might  have 
substituted  f(t)  =  j£7,  a  constant,  and  written  at  once  equa- 
tion (21).  For  in  this  case  we  easily  find  the  required 
integral  : 

m  T    R± 

JEeLdt  =  E~eL. 

1    _«* 
Multiplying  this  by  the  coefficient  je  L  (17)  becomes 

E  — 

~ 


Eeplacing  c  by  —  ->,  we  have 


a  result  identical  with  (21). 


50 


CIRCUITS  CONTAINING 


Here  we  notice  that,  if  the  self-induction  is  zero,  the 
equation  becomes  simply  Ohm's  law ;  that  is,  it  is  the 
self-induction  of  the  circuit  which  prevents  the  current 
from  reaching  its  full  value  immediately  after  the  intro- 
duction of  the  E.  M.  F. 


.02    .04      .06     .08     .10 


.20 


Seconds 


FIG.  12. — CURVE  SHOWING  THE  ESTABLISHMENT  OF  CURRENT  AT  ANY 
TIME  AFTER  THE  INTRODUCTION  OF  AN  E.  M.  F.  INTO  A  CIRCUIT 
WHOSE  RESISTANCE  R  is  .1  OHM  AND  COEFFICIENT  OF  SELF-INDUC- 
TION L  is  .01  HENRY. 

The  increase  of  the  current  with  the  time  is  shown  by 
the  curve  Fig.  12.  This  is  a  logarithmic  curve  similar  to 
that  in  Fig.  9,  with  the  orclinates  measured  downward  from 
the  horizontal  line  O'A,  at  a  distance  above  the  axis  equal 
to  the  maximum  value  /,  of  the  current. 

CASE  III.    HARMONIC    IMPRESSED    E.  M.  F.    IN    A    CIRCUIT 

CONTAINING   A  EESISTANCE  AND   SELF-INDUCTION. 

Let  us  now  suppose  that  in  a  circuit  containing  re- 
sistance and  self-induction  there  is  a  simple  harmonic 
impressed  E.  M.  F.,  that  is  that  the  E.  M.  F.  is  a  sine-func- 
tion of  the  time,  thus  : 


(22)  e  =/(0  = 

Here  ^is  the  amplitude  or  maximum  value  of  the  im- 
pressed E.  M.  F.,  and  GO  is  the  angular  velocity,  equivalent 


RESISTANCE  AND  SELF  INDUCTION.  51 

27T 

to  2?™,  or  -7™-,  where  n  denotes  the  number  of  complete 

periods  per  second,  and  T  the  time  of  one  complete  period. 
The  general  solution  for  the  current,  equation  (17),  is 

i     .Rt    „  **.  -Rt 

(17) 


Substituting  in  (17)  the  value  for  f(t)  in  (22),  the  general 
expression  becomes,  according  to  the  particular  hypothesis 
of  a  sine  E.  M.  F., 

zp    -Rt  p  **1  -  Rt 

Before  integrating  this  equation  we  will  first  obtain  the 
general  integrals 


*  sin  (f3x+6}dx    and        eax  cos  (ftx  +  B)  dx. 
Applying  the  formula  for  integrating  by  parts, 

/  udv  =  uv  —  Iv  du, 
these  integrals  become 


=  sin  (fix  +  6)  .  -        2  fe™  cos  (ftx  +  0)  dx, 
(jK*.*b  B).  eaxdx 

=  cos  (ftx  +  0)  .  ~C-  +  -feax  sin  (ftx  +  ff)dx. 

Eliminating    /  eax  cos  (ftx  -\-  6)  dx  between  these  two  equa- 
tions, we  obtain  as  one  of  the  integrals  sought 

<24)        J  eax  sin  (ftx  +  0)  dx 


52  CIllCVITS  CONTAINING 

Eliminating    /  eax  sin  (fix  -\-  tf)  dx   between   the   same  two 
equations,  we  obtain  in  the  same  way  the  integral 

(25)        f  e™  cos  (fix  +  9)  dx 


?-> 

Keplacing  a  by  ~^,  ft  by  GO,  6  by  0,  and  x  by  £,  in  equa- 

tion (24),  we  have'  the  integration  indicated  in  (23),  and 
equation  (23)  then  becomes 


V  f  7?  \ 

(26)      i  =  — -jp \  yT  sin  <&t  —  o>cos  GO  tj 

- 


oot   4-  ce 


Rt 
L 


This  may  be  written  in  simpler  form  by  the  use  of  the 
trigonometric  formula 

(27)       A  sin  8  +  Bcos  6  =  \/A*  +  B*  sin  (o  +  tan'1  ^Y 

This  formula  is  established  as  follows : 

A  sin  6  -f-  B  cos  B 

B  \ 

=;     COS     »J. 


If  tan  0  =  -j,  then  sin  0  =    .-  ,  and  cos  0  =    . 

'         *  * 


Making  these  substitutions,  we  have 
A  sin  #  +  B  cos  6  =  VA*  +  B*  (cos  0  sin  0  +  sin  0  cos  0) 


which  establishes  the  truth  of  (27). 


RESISTANCE  AND  SELF  INDUCTION.  58 

Keducing  equation  (26)  to  its  simplest  form  by  means 
of  formula  (27),  we  have  from  (26)  the  value  of  the  current 
at  any  instant  of  time. 


(28)    *'= 


DISCUSSION  OF  THE  CURRENT  EQUATION. 

After  a  very  short  time  the  exponential  term  in  this 
equation,  containing  the  arbitrary  constant  of  integration, 
becomes  inappreciably  small,  and  may  be  neglected.  Just 
what  effect  the  exponential  term  has  during  this  short  time 
will  be  considered  later.  The  equation  shows  that,  where 
there  is  an  impressed  sine  electromotive  force  in  a  circuit, 
the  current  is  likewise  a  sine-function  of  the  time,  and  that 
the  current  lags  behind  the  electromotive  force  by  an  angle 

L  GO 

whose  tangent  is  -^-.  If  there  is  no  self-induction  and 
L  =  0,  equation  (28)  becomes 

i  —  -p  sin  co  t, 

which  is  a  direct  result  of  Ohm's  law.  Thus  the  self- 
induction  not  only  causes  the  current  to  lag  behind  the 
impressed  E.  M.  F.,  but  also  diminishes  the  maximum 
value  of  the  current. 

Z 

} 
has  its  maximum  value  /,  and 

E 


When  sin  (  GO  t  —  tan  ~ !  — ^- )  becomes  unity,  the  current 
j  its  maximum  value  7, 

(29)  /= 


+  D  <* 

The  term  "impedance  "has  been  applied  to  the  expression 


*  -\-  U  to2,  the  apparent  resistance  of  a  circuit  contain- 
ing ohmic  resistance  and  self-induction,  and  an  impressed 
sine  electromotive  force. 


54 


CIRCUITS  CONTAINING 


The  equation  (29)  may  be  written 

Maximum  E.  M.  F. 
(30)          Maximum  current  =          lmpedance~ 

Since  virtual  current  =  —-=.  maximum  current,  and  vir- 

1/2 

tual  E.  M.  F.  =  — -  maximum  E.  M.  F.,  see  equation  (12), 
we  may  write 


(31) 


Virtual  current  = 


Virtual  E.M.F. 
Impedance 


Ohmic  Resistance 


LCD 


FIG.  13.— VALUE  OP  IMPEDANCE. 

The  value  of  impedance  is  graphically  represented  in 
Fig.  13.  L  oo  is  sometimes  called  the  inductive  resistance  in 
contradistinction  to  the  oJimic  resistance  R. 

It  has  been  shown  above  that  the  tangent  of  the  angle 

of  lag  is  — D~.     The  angle  of  lag  is  therefore  represented 
by  0  in  Fig.  13. 


Lai 


Effective  e.m.f. 


FIG.  14.— VALUE  OF  IMPRESSED  E.  M.  F. 

The  triangle  may  be  drawn  so  that  the  three  sides  rep- 
resent E.  M.  F.  as  .in   Fig.  14.     Here  E  I  represents  the 


RESISTANCE  AND  SELF  INDUCTION.  55 

E.  M.  F.  necessary  to  overcome  the  ohmic  resistance,  and 
is  in  the  same  direction  as  the  current.  L  GO  Us  at  right 
angles  to  this  and  represents  the  counter  E.  M.  F.  of  self- 
induction.  /  VR*  +  IS  a?  is  the  impressed  electromotive 
force  E.  0  is  the  angle  by  which  the  current  lags  behind 
the  impressed  E.  M.  F. 

Full  discussion  of  the  triangles  of  current  and  E.  Mr  F. 
is  given  in  the  graphical  treatment  of  circuits  with  re- 
sistance and  self-induction,  Chap.  XY. 

It  is  convenient  to  consider  the  impedance  as  a  resist- 
ance, and  the  propriety  of  doing  so  is  shown  by  its  dimen- 
sions, which  are  the  same  as  those  of  resistance,  that  is  a 
velocity  in  the  electromagnetic  system  of  units. 

length 

The  dimensions  of  resistance,  R,  are    ,.        =  velocity. 

time 

The  dimension  of  the  coefficient  of  self-induction,  L,  is 
length.  The  dimension  of  an  angular  velocity  GO  is  —.  —  . 

Therefore  the  dimensions  of  L  GO  are  -p  --  =  velocity,  and 


thus  the  impedance  has  the  same  dimensions  as  a  resistance. 

EXPLANATION  OF  THE  EXPONENTIAL  TERM. 

Let  us  return  to  the  solution  for  current,  equation  (28), 

Rt 

and  consider  the  effect  of  the  exponential  term,  c  e  L  , 
during  the  short  time  after  "  make,"  that  is,  after  the  intro- 
duction into  the  circuit  of  a  simple  harmonic  impressed 
electromotive  force.  The  equation  (28)  for  current  may  be 
written 

_Rt 

(32)  i  =  /sin^+ce     L. 

where     / 


and  fy  =  GO  t  —  tan     --  ; 


56 


CIRCUITS  CONTAINING 


that  is,  /represents  the  maximum  value  and  ^  the  phase  of 
the  current.  The  E.  M.  F.  is  introduced  at  a  time  tr  At 
that  time  the  current  is  zero,  for  the  circuit  is  just  made. 
If  we  call  ^x  the  value  of  ^  when  t  —  tl ,  at  the  introduction 
of  the  E.  M.  F.  equation  (32)  becomes 


Rt, 


(33) 


0  =  Jsin  ip1-}-c  e      L, 


and      c  =  —  le     L   sin 


Substituting  this  value  of  c  in  (32),  the  equation  for  cur- 
rent becomes 


(34) 


= 


—  I  e 


FIG.  15.  —  CURVE  SHOWING  THE  EFFECT  OF  THE    EXPONENTIAL   TERM 
_  m 

c  e          UPON  THE  CURRENT  AT  THE  MAKE,  IN  A  CIRCUIT  WHERE 
L  =  1  HENRY,  R  =  50  OHMS,  GO  =  1000,  ^  =  30°. 

This  equation  may  best  be  explained  by  referring  to 
Fig.  15,  which  represents  the  plot  of  the  equation.  The  par- 
ticular values  assumed  in  this  case  are  L  =  1  henry,  R  —  50 
ohms,  GD  =  1000,  and  0:  =  30°.  The  resultant  current  curve 
III.  is  made  up  of  two  component  parts,  /  sin  ^,  and 


—  le 


T-> 

~        ' 


sin  i/},  ,  which  are  represented  by  the  curves 


RESISTANCE  AND  SELF  INDUCTION.  57 

I.  and  II.  respectively.     Curve  I.  is  a  sine-curve  ana  curve 

II.  a  logarithmic  curve,  the  effect  of  which  upon  the  result- 
ant current  becomes  inappreciable  after  a  very  short  space 
of  time,  in  this  particular  case  after  five  or  ten  periods. 
The  initial  value  of  this  logarithmic  curve  is  equal  and  op- 
posite to  the  value  of  the  ordinate  of  the  component  sine- 
curve  I.  at  the  time  t,  when  the  E.  M.  F.  is  introduced. 
This  is  evident  from  the  equation,  since  the  initial  value  of 
the  logarithmic  curve  is  —  I  sin  i/?l ,  and  the  value  of  the 
sine- curve,  when  t  =  tl ,  is  -f-  /  sin  i/>1 . 

If  another  curve  IV.  is  constructed  so  that  its  ordinates 
represent  the  initial  values  of  the  logarithmic  curve,  when 
the  E.  M.  F.  is  introduced  at  different  points  in  the  period, 
it  is  seen  to  be  simply  a  .sine-curve,  corresponding  with  the 
component  curve  I.  but  reversed,  or,  what  is  the  same 
thing,  differing  from  it  by  180°  in  phase. 

To  conclude,  we  see  that  the  effect  of  the  exponential 
term  in  the  equation  is  a  maximum  if  the  E.  M.  F.  is  intro- 
duced at  that  point  of  its  phase  at  which  the  current  has 
its  maximum  value  when  everything  has  reached  its  per- 
manent state ;  this  term  has  no  effect  if  the  E.  M.  F.  is 
introduced  at  that  point  of  its  phase  at  which  the  current 
has  its  zero  value  when  everything  has  reached  its  per- 
manent state. 

CASE  IV. — PERIODIC  E.  M.  F.  WHICH  is  NOT  HARMONIC,  IN 
CIRCUITS  CONTAINING  KESISTANCE  AND  SELF-INDUCTION. 

In  Case  III.  the  solution  was  given  for  a  circuit  contain- 
ing an  impressed  E.  M.  F.  which  was  a  simple  sine-function- 
of  the  time.     Now  let  us  suppose  that  the  E.  M.  F.  does 
not  follow  a  simple  sine  law,  but  that  it  is  the  sum  of  two 
components  each  following  a  sine  law,  that  is, 

(35)  e  —  E^mGot  +  E,  sin  (boot  +  0). 


58  CIRCUITS  CONTAINING 

Substituting  in  the  general  expression  for  current  (17) 
this  value  for/  (t),  we  have 


Rt  Rt 


(36)  t-^e  *:j.i*ju*ia 


Le       J  ( 

Performing  the  indicated  integrations  by  use   of  the 
formula  of  integration  (24),  we  have 


_  R    • 

(37) 


ZF 

Tfl  .  f      Tf 

+  — /7?a    " r    -j   j  sin  (boot -}-&)  —  boo  cos  (boat-}-  6)  v 

L       #* 


Simplifying  by  formula  (27)  this  becomes 
(38)          i  = 


By  this  equation  it  is  seen  that  each  simple  sine  impressed 
E.  M.  F.  gives  rise  to  a  simple  sine  term  in  the  resulting 
current  equation.  The  result  may  therefore  easily  be 
extended,  and  we  may  say  that,  if  there  are  n  simple  sine 
impressed  E.  M.  E.'s  of  the  form  E  sin  (b  GO  t  +  6),  where 


i 

RESISTANCE  AND  SELF  INDUCTION.  59 

E,  b,  and  0  have  different  values  in  each  component  term, 
the  current  equation  will  be  the  sum  of  n  terms  of  the  form 


E 


(  ^Lbco] 

sin   \  b  GO  t  +  6  —  tan      -^—  V 

3  &'  a?'          L  R     J 


_Rt 

plus  the  term  c  e    L  containing  the  arbitrary  constant. 

Here  E,  b,  and  9  have  the  same  values  in  each  term  as 
they  do  in  the  corresponding  term  of  the  impressed  E.  M.  F* 

Expressing  the  current  by  a  summation,  we  have 


(39)t  =  *>  E-=  sin  -I  b  GO  t  +  0  - 

-V&    L*VGO*     \ 


Rt 


when  the  impressed  E.  M.  F.  is 

(40)  e  - 

In  these  sums  E,  b,  and  6  may  have  n  values,  but  they 
must  be  the  same  values  in  each  sum,  giving  rise  to  the 
same  number  of  terms  in  each. 

It  was  first  shown  by  Fourier  that  such  a  sum  of  simple 
sine  terms  as  that  represented  in  equation  (39)  may  express 
any  single-valued  function  whatever,  and  thus  we  see  that 
the  equation  expresses  the  most  general  case  of  a  current 
flowing  in  a  circuit  with  resistance  and  self-induction,  and 
may  represent  the  current  caused  by  any  E.  M.  F.  what- 
soever, 

The  consideration  of  this  most  general  expression  for 
the  current  will  be  deferred  until  the  case  has  been  taken 
up  where  the  circuit  not  only  contains  resistance  and  self- 
induction,  but  also  a  condenser. 


CHAPTER  IV. 

INTRODUCTORY  TO  CIRCUITS  CONTAINING  RESISTANCE 
AND   CAPACITY. 

CONTENTS:— Plan  to  be  followed.  Charge.  Law  of  force.  Unit  charge. 
Work  in  moving  a  charge.  Potential.  Capacity.  Energy  of  charge. 
Condenser,— energy  of  and  capacity  of.  Capacity  of  parallel  plates; 
of  continuous  conductor.  Equation  of  energy,  in  terms  of  i\  in  terms 
of  q.  Equation  of  E.  M.  F.'s. 

IN  the  first  chapter  the  fundamental  principles  neces- 
sary to  lead  up  to  the  derivation  of  the  equation  of  energy 
for  circuits  containing  resistance  and  self-induction  only 
were  given  ;  then  followed,  in  the  third  chapter,  the  solu- 
tion of  this  differential  equation,  which  enabled  us  to  ascer- 
tain the  current  flowing  in  the  circuit  at  anytime.  Follow- 
ing a  similar  plan,  there  will  be  given  in  this  chapter  the 
necessary  fundamental  principles  which  lead  up  to  the 
derivation  of  the  differential  equation  of  energy  for  circuits 
containing  resistance  and  capacity,  and  in  the  following 
chapter  the  general  solution  of  this  differential  equation 
and  its  application  to  various  particular  cases. 

LAW  OF  FORCE. 

Every  one  is  familiar  with  the  fact  that  bodies  may  be 
charged  with  electricity,  and  that  two  like  charges  repel 
and  two  unlike  charges  attract  one  another.  It  was  found 
from  experiment  by  Coulomb  that  if  we  have  two  charges, 
each  concentrated  at  a  point,  the  force  of  attraction  or 

60 


CIRCUITS  CONTAINING  RESISTANCE  AND  CAPACITY.   61 

repulsion  between  them  varies  directly  with  the  product  of 
the  two  charges  and  inversely  as  the  square  of  the  distance 
between  the  two  points,  that  is, 


where  q  and  q'  represent  the  quantities  of  the  charges,  r  the 
distance,  and  F  the  force  between  them.  When  the  quantities 
considered  have  the  same  sign,  the  product  q  q'  is  positive, 
and  therefore  a  force  of  repulsion  has  a  positive  sign. 
Similarly  a  force  of  attraction  has  a  negative  sign. 

If  the  distance  between  these  points  is  unity,  the 
charges  being  equal,  and  if  the  force  between  them  is  a 
unit  force,  each  charge  is  called  a  unit  charge.  The  defini- 
tion of  the  electrostatic  unit*  of  quantity  of  electricity,  in 
the  C.  G.  S.  system,  is  then:  that  quantity  which,  when 
placed  at  a  distance  of  one  centimeter  from  an  equal  quan- 
tity (in  a  medium  whose  specific  inductive  capacity  is  unity 
—  that  is,  in  air  or  vacuo),  repels  it  with  the  force  of  one 
dyne.  Where  these  units  are  used,  and  the  medium  is 
a  vacuum,  the  law  of  force  may  be  written 


Where  the  medium  is  not  a  vacuum,  the  force  is  found  to 
be  less  and  equal  to 


K  r 


where  K  is  a  constant  quantity  called  the  specific  inductive 
capacity  of  the  medium. 

POTENTIAL. 

Since  there  exists  a  force  between  two  charges  of  elec- 
tricity, mechanical  work  is  done  if  either  is  moved  so  as  to 
change  the  distance  between  them.  The  work  done  in 


62  INTRODUCTORY  TO   CIRCUITS 

moving  any  body  against  a  uniform  force  is  equal  to  the 
product  of  the  force  and  the  distance  through  which  the 
body  is  moved  against  that  force.  The  force  between  the 

electrical  charges  q  and  q'  is  -^-f-.     If  they  be  moved  in  any 

direction  whatsoever,  so  that  the  distance  r  between  them 
is  changed  to  r  -f-  dr,  the  work  done  in  moving  them  is  the 

product  of  the  force  — 5-  and  the  change  in  the  distance  drt 

since  the  force  may  be  considered  constant  throughout  the 
small  distance  dr.  Therefore  the  work  is- 


FIG.  16.—  WORK  DONE  IN  MOVING  A  CHARGED  BODY. 

Suppose  a  charge  g  is  situated  at  the  point  A  (Fig.  16), 
and  a  charge  q'  is  moved  from  the  point  P1  to  P2.  The 
work  done  by  the  electric  force  in  moving  the  charge  is 


or,  the  work  done  against  the  electric  force  isqq'(  —  ---  J. 

It  is  seen  that  the  work  done  in  moving  a  charge  from 
one  point  to  another  is  independent  of  the  path  by  which  it 
is  moved,  and  simply  depends  on  the  initial  and  final  dis- 
tances between  the  charge  q  and  q'.  If  the  distance  rl  is 
infinite  (meaning  that  the  charge  q'  is  carried  from  an  in- 


. 
CONTAINING  RESISTANCE  AND   CAPACITY.  63 

finite  distance  to  a  point  at  a  distance  r2),  the  work  done 
against  the  electric  force  becomes  simply 


If  q'  is  unity  and  a  unit  charge  is  moved,  the  work  becomes 


It  is  seen  that  each  point  in  the  region  surrounding  an 
electric  charge  possesses  a  certain  characteristic  which 
determines  the  amount  of  work  done  in  bringing  a  charge 
from  infinity  to  that  point.  This  characteristic  of  the 
point  has  been  called  its  potential.  The  potential  V  at  a, 
point  is  therefore  defined  as  the  work  done  in  moving  a 
unit  positive  charge  from  an  infinite  distance  to  that  point  ; 

thus,   V  •=•  —  .     This  potential  is  positive  when  the  work 

is  positive,   that  is,  when  work  is  done,  in  moving  the 
charge,  by  some  agent  external  to  the  system. 

The  potential  at  a  point  due  to  a  number  of  charges, 
each  concentrated  at  a  point,  is  the  sum  of  the  potentials 
at  that  point  due  to  each  charge  independently  ;  thus, 

"" 

If  there  is  a  charge  distributed  upon  any  surface  and 
dq  is  the  charge  upon  an  element  of  that  surface,  the  poten- 
tial at  any  point  due  to  this  charged  surface  is  equal  to  the 
sum  of  the  potentials  due  to  each  elemental  charge  ;  that  is, 


The  potential  at  every  point  of  a  good  conductor  is  the 
same,  since  the  electricity  will  so  distribute  itself  on 
the  body  that  no  work  would  be  done  by  transferring  a 


64  INTRODUCTORY  TO  CIRCUITS 

charge  from  one  point  of  the  conductor  to  another  point  of 
it.  This  potential  Fis  called  the  potential  at  the  conductor, 
and  the  conductor  is  said  to  be  at  potential  V.  The  poten- 
tial at  a  conductor  may  be  due  partly  or  wholly  to  the 
charge  on  the  conductor  itself. 

CAPACITY  OF  A  CONDUCTOR. 

The  potential  of  a  charged  body  is  directly  proportional 
to  its  charge,  that  is,  Foe  q,  or  q  =  .C  V,  where  C  is  some 
constant;  for,  suppose  the  body  possesses  a  unit  charge 
and  its  potential  is  V\  a  second  unit  charge  brought  from 
infinity  to  the  body  doubles  its  original  charge.  The  po- 
tential is  then  2  J7,  for  the  potential  is  the  work  done  in 
bringing  a  unit  charge  from  infinity  to  the  point,  and  the 
work  in  bringing  a  unit  charge  to  a  body  with  a  quantity 
2g  is  twice  the  work  in  bringing  a  unit  charge  to  a  body 
with  a  quantity  q.  We  thus  see  that  q  is  proportional  to 
V,  and  is  consequently  equal  to  V  multiplied  by  some  con- 
stant, that  is, 

(41)  q=CV. 

If  a  body  is  charged  to  a  unit  potential  and  the  quan- 
tity is  q, 

q=C. 

C  is  therefore  defined  as  the  quantity  of  electricity  upon 
a  body  when  at  a  unit  potential.  This  is  called  the  capacity 
of  the  conductor.  The  capacity  depends  upon  the  size  and 
geometrical  form  of  the  conductor  and  the  specific  inductive 
capacity  of  the  surrounding  medium. 

ENERGY  OF  A  CHARGED   CONDUCTOR. 

Suppose  a  body  is  charged  with  a  quantity  of  electricity 
q,  and  is  at  a  potential  V.  The  work  done  in  bringing  a 
unit  quantity  of  electricity  from  an  infinite  distance  up  to 
the  body  is  Fby  definition.  (This  is  provided  q  is  so  large 


. 
CONTAINING  RESISTANCE  AND  CAPACITY.  65 

in  comparison  with  a  unit  quantity  that  its  potential  is  not 
appreciably  altered  by  the  addition  of  the  unit  quantity.) 
If,  under  the  same  conditions,  we  bring  up,  not  a  unit 
quantity,  but  a  quantity  dq,  the  work  done  is  Vdg,  and  this 
represents  the  increment  of  the  energy  of  the  charge  q. 
That  is, 


(42)  dW  =  Fdq. 

Eeferring  to  equation  (41),  we  may  always  replace  V  by 
its  equal  ^  or  dq  by  its  equal  C  dV,  and  obtain  the  equa- 
tions 


and 

dW=  CVdV. 

The  integrals  of  these  equations,  taken  between  the 
limits  zero  and  q,  and  zero  and  F,  respectively,  are 


Since  q  —  C  F,  each  of  these  equations  may  be  written 
(43)  W=%qV. 

Here  JFis  the  potential  energy  possessed  by  the  charged 
body,  as  the  limits  of  integration  were  taken  from  zero 
charge  to  charge  q,  and  from  zero  potential  to  potential  V. 

CAPACITY  AND  ENERGY  OF  A   CONDENSER. 

A  condenser  is  a  device  for  increasing  the  capacity  of  a 
conductor  by  bringing  it  near  another  similar  conductor, 
which  is  separated  from  it  by  any  non-conducting  medium 
or  dielectric.  This  dielectric  will  be  considered  to  be  a 


66  INTRODUCTORY  TO   CIRCUITS 

perfect  non-conductor  ;  that  is,  the  condenser  is  not  leaky. 
A.  condenser  usually  consists  of  two  sets  of  parallel  plates 
alternately  connected,  and  separated  by  a  distance  very 
small  as  compared  with  the  dimensions  of  the  plates.  The 
two  sets  of  plates  are  usually  called  simply  the  two  plates 
of  the  condenser.  When  the  condenser  is  charged,  the  two 
plates  have  equal  quantities  of  electricity  upon  them,  but 
of  the  opposite  sign. 

The  total  energy  of  a  charged  condenser  may  readily  be 
found  by  taking  the  algebraic  sum  of  the  energies  of  the 
charge  on  each  plate,  as  given  by  the  equation  (43). 

If  the  plates  of  a  condenser  have  charges  -|-  q  and  —  q 
at  potentials  Vl  and  F2  ,  respectively,  the  total  energy  is 


(44) 


that  is,  the  energy  of  a  charged  condenser  is  equal  to  one- 
half  the  product  of  the  charge  of  one  of  the  plates  and  the 
difference  of  potential  between  the  plates.  If  this  differ- 
ence of  potential  between  the  plates  is  simply  V,  the  ex- 
pression for  the  energy  of  a  charged  condenser  is 


(45) 


The  capacity  C  of  a  condenser  is  the  quantity  of  elec- 
tricity on  one  plate  when  there  is  a  unit  difference  of  poten- 
tial between  the  plates  ;  and  when  there  is  a  difference  of 
potential  Fthe  charge  is 

(46)  q=CV. 

It  can  be  shown  that  the  capacity  of  a  condenser,  com- 
posed of  parallel  plates  of  equal  area,  whose  distance  apart 
is  small  as  compared  with  the  dimensions  of  the  plates,  is 
directly  proportional  to  the  area  of  the  plates,  and  inversely 


CONTAINING  RESISTANCE  AND   CAPACITY.  07 

proportional  to  the  distance  between  them,  and  that  the 
capacity  is 

<47>  :''  -' 


where  A  is  the  area  of  each  plate  and  d  the  distance  be- 
tween the  plates. 

As  the  plates  of  a  condenser  approach  nearer  and  nearer 
together,  the  capacity  C  becomes  larger  and  larger.  In  the 
limit,  when  the  plates  come  into  contact,  the  capacity  be- 
comes infinite,  which  means  that,  no  matter  how  much  one 
plate  is  charged,  there  can  exist  no  difference  of  potential 
between  them.  If,  then,  a  circuit  is  a  continuous  con- 
ductor and  has  no  condenser  in  it,  it  may  be  said  to  have 
a  condenser  of  infinite  capacity  in  series  with  it. 

By  combining  equations  (45)  and  (46)  the  energy  of  the 
charge  of  the  condenser  may  be  expressed  in  terms  of  the 
capacity  and  the  potential  V,  or  in  terms  of  the  capacity 
and  the  charge  q.  Thus, 

The  increment  of  the  energy  d  JF,  as  the  potential  and 
charge  vary  simultaneously,  is 

qdq 


<49) 


C 


THE   EQUATION   OF   ENEEGY. 

We  can  now  write  the  equation  of  energy  for  an  electric 
circuit  having  a  resistance  J?,  and  having  in  series  with 
that  resistance  a  condenser  of  capacity  C. 

The  total  energy  given  to  the  circuit  by  the  source  of 
E.  M.  F.  is  eidt;  and  that  part  of  the  energy  used  in  heat- 
ing the  conductor  in  the  time  dt  is  R  ?  dt,  as  shown  in 
•equations  (5)  and  (4).  The  amount  of  energy  required  in 

dW 
the  time  dt  to  change  the  charge  of  the  condenser  is  —-  dt. 


68  INTRODUCTORY  TO   CIRCUITS 

Since,  under  the  conditions  supposed,  these  two  are  the 
only  ways  in  which  the  energy  imparted  by  the  source  is 
used,  we  have  the  equation  of  energy, 

dW 
(50)  eidt  =  R i* dt  +  -=-  dt. 


We  have  -seen  that  dW=  —TT  (equation  49)  ;  therefore 


(51)  eidt  =  Ri^dt+--dt. 

\j    Cut 

When  a  current  i  flows  into  a  condenser  for  a  time  dt, 
the  quantity  which  flows  during  this  time  is  i  dt,  but  this 
is  ih  e  increment  dq  of  the  charge  of  the  condenser,  that  is, 

dq  =  i  dt  ; 
hence 

(52)  q  =  f'idt. 

Substituting  these  values^of  q  and  i  in  equation  (51)  we 
may  write  the  equation  of  energy  in  two  forms,  in  terms  of 
i  or  in  terms  of  g,  thus  : 

idt  fidt 

(53)  eidt  =  Ri*dt-\  ---    --  J 


Dividing  (53)  through  by  idt  and  (54)  by  its  equal  dq, 
we  have 

fidt 
(55)  €  =  Ki  +  tL-Q--> 

ft*  7?^_i_  q 

(56;  e  =  E-    +  ~' 


CONTAINING  RESISTANCE  AND   CAPACITY.  69 

These  are  equations  of  electromotive  forces,  where  e  is 

the  impressed  E.  M.  F.  of  the  source,  R  i  or  R  -rr   the 

E.  M.  F.  necessary  to  overcome  the  ohmic  resistance,  and 

Jidt       q 

—  77—  =  -rj  =  V,  the  E.  M.  F.  necessary  to   oppose   the 
o         o 

E.  M.  F.  of  the  condenser. 

When  C  is  infinite,  that  is,  as  explained  above,  when  the 
plates  of  the  condenser  come  into  contact,  we  have  a  circuit 
with  resistance  only,  in  which  'case  equation  (55)  gives 


which  is  Ohm's  law. 


CHAPTER  V. 

CIRCUITS    CONTAINING    RESISTANCE   AND   CAPACITY. 

CONTENTS. — Equation  of  E.  M.  F.'s.  Differential  equation  in  linear  form. 
Criterion  of  integrability.  General  solution  when  e  =f(t). 

Case  I.  Discharge.  Quantity  and  current  *rom  general  solution, — from 
differential  equations. 

Case  II.  Charge.  Quantity  and  current  from  general  solution, — from  dif- 
ferential equations. 

Case  III.  Simple  harmonic  E.  M.  F.  Quantity  and  current  from  general 
solution.  Discussion. 

Case  IV.  Any  periodic  E.  M.  F. 

IN  the  previous  chapter  the  equation  of  energy  for 
a  circuit  containing  ohmic  resistance  and  capacity  was 
derived,  and,  by  dividing  the  equation  of  energy  through 
by  i  dt  or  dq,  it  was  found  that  the  equation  of  electro- 
motive forces  thus  obtained  may  be  expressed  in  terms  of 
current,  i9  or  charge,  g,  thus : 

(55)  .  -          e  =  si 

(56)  e  =  Rd£+^.  '••          ~      ' 

Differentiating  (55),  to  free  it  from  the  integral  sign 
and  transposing,  the  two  equations  may  be  written  : 

(57)  Cde  -  -  R  Cdi  —  idt  =  0. 

(58)  Ode-tf  Cdq  +  (eC-q)dt  =  0. 

70 


RESISTANCE  AND  CAPACITY. 


71 


Each  of  these  equations  is  a  differential  equation  of  the 
first  order  with  three  variables,  e,  i,  and  t,  and  e,  g,  and  t, 
respectively,  of  the  form 

P  dx  -f  Q  dy  +  S  dz  =  0. 

If  there  exists  a  single  integral  equation  of  which  this 
is  the  exact  differential,  the  condition  of  integrability  * 


dQ     d8 


d8     dP 


dP 


-^  =  0 

dx  ) 


nmst  be  satisfied. 

Applying  this  criterion  of  integrability  to  the  equations 


FIG.  17.— CIRCUIT  HAVING  OIIMIC  RESISTANCE  AND  CAPACITY. 

(57)  and  (58),  it  is  found  that  the  condition  is  not  satisfied 
by  either  equation.  No  single  equation  exists,  therefore, 
of  which  (57)  or  (58)  is  an  exact  differential. 

But,  as  was  previously  stated,  we  know  that  the  electro- 
motive force  e  may  always  be  expressed  as  a  single -valued 
function  of  the  time,  since  it  must  have  some  one  value  at 
each  point  of  time,  and  we  have 


(59) 


«=/(<). 


where  /  is  an  arbitrary  single-valued  function.     By  differ- 
entiation (59)  becomes 


(CO) 


de 

'     =/'('> 


*  See  Johnson's  Differential  Equations,  p.  270, 


72  CIRCUITS  CONTAINING 

Equations  (57)  and  (58)  may  now  be  written  in  the  linear 
form  thus : 


The  solutions  of  these  linear  equations  *  are 
(63)  »  =  - 


7   /*  .  JL  __L 

/    ^.BC -,.v  ,.   .  #a 

•J  e    /• 


_  __ 

(64)  V=-z-J  e     'Cf(^dt+c,e    ' 

The  integrals  here  expressed  cannot  be  found  unless  we 
know  in  what  particular  way  the  electromotive  force  varies 
with  the  time.  When  we  know  this,  these  equations  will 
give  the  values  of  the  current  and  charge  at  any  time, 
provided  the  integral  sought  can  be  obtained.  We  will 
now  assume  several  ways  in  which  the  E.  M.  F.  varies 
with  the  time,  which  will  allow  the  integration  to  be  easily 
performed. 

CASE  I.    DISCHARGE  OF  A  CONDENSER. 

Suppose  that  a  constant  source  of  E.  M.  F.,  E,  has  been 
acting  upon  a  circuit  containing  in  series  a  resistance,  and 
a  condenser  with  capacity  (7,  until  everything  has  reached 
its  steady  state.  No  current  will  be  flowing,  and  the  con- 
denser will  be  charged  with  a  quantity  Q,  and  have  a  dif- 
ference of  potential  E  at  its  terminals.  Now  suddenly 
remove  the  source  of  E.  M.  F.  from  the  circuit  and  suppose 
its  resistance  then  is  J?.  The  condenser  will  immediately 

*  See  Johnston's  Differential  Equations,  p.  31. 


RESISTANCE  AND  CAPACITY.  73 

begin  to  discharge  through  the  conductor,  and  we  wish  to 
find  the  value  of  the  charge  q  and  current  i  at  any  time 
after  the  discharge  begins. 

When  the  E.  M.  F.  was  removed  from  the  circuit  the 
impressed  E.  M.  F.,  e  =f(t),  became  equal  to  zero  at  every 
point  of  time  after  the  removal ;  hence,  substituting 

(65)  e  =f(f)  =  0 

in  the  general  equations  (63)  and  (64),  we  find  that  the  in- 
tegral vanishes,  and  we  have  the  immediate  results, 


t 

R~C 


t 

RC 


The  arbitrary  constants  c,  and  c2  are  determined  by  the 
initial  conditions.  If  the  charge  is  Q  when  the  time  is 
zero,  the  charge  equation  becomes 

(66)  g=Qe~jrG- 

and  since  dq  =  i  dt,  the  current  equation  becomes 

(67)     r-      '     i=--         - 


If,  instead  of  substituting  e  =  f(t)  =  0  in  the  general 
solutions  (63)  and  (64),  we  had  substituted  in  the  differ- 
ential equations  (61)  and  (62),  it  is  seen  that  the  second 
member  of  each  becomes  zero,  and  that  the  solutions  are 
merely  the  "complementary  functions,"  namely,  the  terms 
in  the  general  solutions  containing  the  arbitrary  constants, 
as  pointed  out. 

It  may  be  of  interest  to  derive  the  solution  directly  from 
the  differential  equations,  since  the  variables  easily  admit 
of  separation. 


74  CIRCUITS  CONTAINING 

Equation  (62),  when/(Q  =  0,  is 

/VXV  XV 

T  =  0. 

J 

dt 


dt  "•  J?  i 

dg 
Hence     —  =  — 


and 


t 


or  <7  =  ce         , 

which  is  identical  with  (67). 

In  Fig.  18  is  shown  a  curve  of  discharge  of  a  condenser 
for  a  particular  case.  The  rapidity  of  discharge  is  shown 
by  the  value  of  the  time-constant  T,  which  gives  the  time 
in  which  the  charge  of  the  condenser  is  reduced  to  one  eth 
of  its  initial  value. 

T  =  R  G  =  100  X  109  X  4  X  10-16  =  .0004  seconds. 


FIG.   18. — CURVE   SHOWING   DISCHARGE  OF  A  CONDENSER  WHOSE  CA- 
PACITY C  —  4  MICROFARADS,  THROUGH  A  RESISTANCE  R  =  100  OHMS. 

CASE  II.    CHARGE  OF  A  CONDENSER. 

Suppose  that  a  constant  source  of  E.  M.  F.,  E,  is  sud- 
denly introduced  into  a  circuit,  and  that  the  resistance 
when  it  is  introduced  is  R,  the  capacity  of  the  condenser 
in  series  with  the  resistance  being  C.  The  values  of  the 


RESISTANCE  AND   CAPACITY.  75 

current  i  and  charge  q  at  any  time  after  the  introduction 
of  the  E.  M.  F.  will  be  given  by  equations  (63)  and  (64)  if 
we  suppose 

(68)  e  =f(t)  =  E,  a  constant, 

de 
and  consequently  -r  =f'(t)  =  Q. 

Substituting  these  values,  (63)  and  (64)  become 

(69)  i  =  c,  e 


t 

RC 


(70) 


t 

R~C 


Determining  the  constants  of  integration  c,  and  c2  by  the 
condition  that  there  was  no  charge  in  the  condenser  when 
t  =  0,  we  have 

c2  =  -  CE. 

But  since  C£  '=  Q,  the  final  charge  of  the  condenser  when 
everything  has  reached  its  steady  state,  (70)  becomes 


~ 
(71)  g=Ql-e       c 

and  by  the  relation  dq  =  idt  equation  (69)  becomes 


(72)  <  =  3f-** 

It  is  noticeable  that  the  equations  for  the  current  (67) 
and  (72)  are  identical  in  the  case  of  charge  and  discharge 
of  a  condenser,  except  that  the  sign  of  i,  i.e.,  the  direction 
of  the  current,  is  reversed. 

Equation  (71)  may  easily  be  derived  from  the  differ- 
ential equation  (62)  directly,  upon  substituting  f(t)  =  Ey 
as  the  variables  easily  admit  of  separation  ;  thus, 

dq_         q_       E 
dt  -~RC~  B 


76 


CIRCUITS  CONTAINING 


may  be  written 

dg 

dt 

q-GE 

EG' 

and 

(q  -  CE)  _ 

t 

c, 

~  EC* 

TT«,,, 

SI  UT     I 

t 

~  RC 

wliich  is  identical  with  (70). 

The  curve  representing  the  charge  of  a  condenser  is 
shown  in  Fig.  19.  The  time-constant  E  C  =  .0004.  The 
final  charge  is 

Q  =  C  V  =  4  X  10-15  X  200  X  108  X  10  =  .0008  coulombs. 


Seconds 


FIG.  19.— CURVE  SHOWING  THE  CHARGE  OF  A  CONDENSER  WHOSE  CA- 
PACITY C—  4  MICROFARADS  WHEN  SUBJECTED  TO  A  DIFFERENCE  OF 
POTENTIAL  OF  200  VOLTS  THROUGH  A  RESISTANCE  OF  100  OHMS. 

The  curve  of  discharge  for  the  same  condenser  under 
the  same  conditions  was  given  in  Fig.  18. 


CASE  III.    ELECTKOMOTIVE  FORCE  A  SIMPLE  HARMONIC 
FUNCTION  OF  THE  TIME. 

Let  us  now  suppose  the  impressed  E.  M.  F.  to  be  a 
simple  harmonic  function  of  the  time,  as  in  Case  III.,  Chap. 


RESISTANCE  AND   CAPACITY.  77 

III.,  in  the  discussion  of  circuits  containing  resistance  and 
self-induction  ;  that  is, 

(73)  e=/(Q  =  JFsin  cot, 

where    E  is   the    amplitude,    or   maximum    value   of    the 
E.  M.  F.,  and  GO  the  angular  velocity.     By  differentiation, 

de 

=ff  (t)  =  EGO  cos  cot. 


Substituting  these  valuos  in  the  general  equations  (63)  and 
(64),  we  obtain 


(74) 


(75) 


RCRC  RC 


RC 


These  integrals  may  be  found  by  the  formulae  of  reduc- 
tion, obtained  by  integrating  by  parts,  given  in  equations 
(25)  and  (24). 

Applying  these  formulae  of  reduction  to  equations  (74) 
and  (75),  they  become 

(76)     i  =  * 


C*  E  E      j    1  )  __ 

(77)    q  =  ,  *™  "t-™  COS 


These  equations  (76)  and  (77)  may  be  simplified  by  the 
trigonometric  formula 

(27)       A  sin  0  +  B  cos  0  =  VA*  +  B*  sinj  B  +  tan'1  j  [. 


78  CIRCUITS  CONTAINING 

By  the  application  of  this  formula  to  (76)  and  (77)  we 
have  the  complete  solutions  of  the  differential  equations, 
namely, 

E  (  I     I  -  ± 

i  =  — / -  sinSc^-f.tan-1-^-^—  \-\-  GI  e 

1\j  _/X  GO  i  * 

. 


and 

v  __L 

RC 


9  = 

GO 


The  last  equation  is  equivalent  to 
(79)    q  =  -       ,^-     =  cos 


These  equations  (78)  and  (79)  are  the  complete  solutions, 
expressed  in  their  simplest  forms.  It  will  be  noticed  that 
the  differential  of  (79)  is  (78),  according  to  the  relation  dg 
=  i  dt.  It  was  not  necessary  .to  carry  both  equations 
through  together,  as  one  may  be  directly  derived  from  the 
other  by  integration  or  differentiation.  It  is  thought  it 
may  add  interest  to  the  case  if  we  have  the  two  to  compare, 
so  that  any  differences  that  exist  become  more  apparent. 

After  a  very  short  time  the  last  term  of  each  of  these 
equations,  containing  the  arbitrary  constant  of  integration, 
becomes  inappreciably  small  and  may  be  neglected.  Then 
it  is  seen  that  the  current  and  charge  are  both  harmonic 
functions  of  the  time  ;  but  the  current,  instead  of  lagging 
behind  the  impressed  E.  M.  F.,  as  it  did  in  the  case  where 
there  was  self-induction  in  the  circuit,  advances  ahead  of  it 

by  an  angle  whose  tangent  is  77-^ — .     When  the  capacity 

O  ±1  GO 

C  is  infinite  (and  there  is  no  condenser  in  the  circuit,  as  ex- 


EESISTANGE  AND   CAPACITY.  79 

plained  on  page  67)  the  tangent  ~TD  —  is  zero,  and  the  cur- 


rent is  in  phase  with  the  E.  M.  F.     When  the  condenser 
is   short-circuited    so   that   the   resistance    is   negligible, 

T^-D  —  becomes  very  large   and  the  angle  of   advance  is 

L>  ll>  GO 

nearly  90°. 

The  equation  of  the  current  then  becomes 


(80)  i=  CEGosin(Got  +  90°), 
and  of  charge 

(81)  q  =  —  Q  cos  (GO  t  +  90°)  ; 

and  the  charge  will  always  be  a  maximum  when  the  current 
is  zero  and  vice  versa,  as  the  cosine  is  a  maximum  when  the 
sine  is  zero. 

When  the  sin  j  oot  -f-  tan~'  ^  „      j-  becomes  unity  the 
current  has  its  maximum  value  I,  and 

(82)  /=  E 


The  radical  A  /  R*  _|  --  _  _  is  the  apparent  resistance  of 

y  C      Ge9 

the  circuit  ;   and,  upon  comparing  with  equation  (29),  we 


see  that  it  corresponds  to  the  radical  \ff  -\-  D  &/,  which 
has  been  called  the  "  impedance"  of  the  circuit,  in  the  case 
where  there  is  self-induction  and  resistance  only. 

CASE  IV.    ANY  PERIODIC  ELECTROMOTIVE  FORCE  WHICH  is 
NOT  HARMONIC. 

If  the  impressed  electromotive  force  is  any  periodic 
function  whatsoever  of  the  time,  then  —  as  was  mentioned 


80   CIRCUITS  CONTAINING  RESISTANCE  AND   CAPACITY. 

in  the  discussion  of  circuits  containing  self-induction  —  this 
E.  M.  F.  may  be  expressed,  according  to  a  theorem  due  to 
Fourier,  as  the  sum  of  terms  of  the  form 


Thus, 

(83)  e  =          Earn  (b  GO  t  +  0) 

E,  6,  e. 

may  represent  any  electromotive  force  whatsoever,  where 
E,  b,  and  0  have  n  different  values  corresponding  to  n  terms 
of  the  sum.  As  was  previously  shown  in  the  case  of  self- 
induction,  each  term  of  the  E.  M.  F.  impressed  gives  rise 
to  a  corresponding  term  in  the  resultant  current  equation 
of  the  form 

rr 


where  E,  b,  and  0  have  values  equal  to  their  values  in  the 
corresponding  term  of  the  E.  M.  F.  equation. 

The  expression  for  current,  then,  when  (83)  is  the  im- 
pressed E.  M.  F.,  is 

rrr 

(84)     i  = 


This  gives  the  general  solution  for  the  current  in  a 
simple  circuit  containing  resistance  and  capacity,  and  any 
impressed  E.  M.  F.  The  discussion  of  this  general  solution 
will  be  deferred  until  circuits  containing  resistance,  self- 
induction,  and  capacity  have  been  considered. 


CHAPTER  VI. 

CIRCUITS  CONTAINING  RESISTANCE,   SELF  INDUCTION, 
AND  CAPACITY.     GENERAL  SOLUTION. 

CONTENTS. — Equation  of  energy  in  terms  of  e,  i,  and  t ;  in  terms  of  e,  q,  and 
t.  Equation  of  E.  M.  F.'s  in  terms  of  e,  i,  and  t ;  in  terms  of  e,  q,  and  t. 
Equations  transformed  for  solving  in  terms  of  i  and  t ;  in  terms  of 
q  and  t.  Complete  solution  for  i  in  terms  of  t  ;  complete  solution  for  q 
in  terms  of  t.  Four  cases  will  be  considered:  I.  e  =f(t)  =  0;  II. 
e  =f(t)  =  E;  III.e=f(t)  =  Esin  cot;  IV.  e  =f(t)  =  2E  sin  (boot  +  6). 

IN  the  preceding  chapters  the  formation  of  the  differ- 
ential equations  for  circuits  containing  resistance  and  self- 
induction  alone,  and  resistance  and  capacity  alone,  has  been 
discussed,  and  the  solution  of  these  differential  equations 
obtained  and  discussed  for  these  two  particular  cases.  It 
is  now  proposed  to  consider  a  circuit  containing  all  three, 
resistance,  self-induction,  and  capacity,  in  series,  and  in  the 
present  chapter  to  derive  from  the  differential  equations 
two  general  solutions  which  express,  respectively,  the  cur- 
rent flowing  in  the  circuit  and  the  charge  of  electricity  in 
the  condenser,  at  any  moment,  when  the  circuit  is  sub- 
jected to  any  impressed  electromotive  force  whatsoever. 
The  succeeding  five  chapters  of  Part  I  will  then  be  devoted 
to  a  discussion  of  these  general  equations,  now  to  be  ob- 
tained, and  their  application  to  various  particular  cases  of 
impressed  electromotive  forces. 

The  differential  equation  of  energy  for  a  circuit  contain- 
ing all  three,  resistance,  self-induction,  and  capacity,  may 

81 


82  CIRCUITS  CONTAINING 

be  written  at  once,  since  we  have  already  derived  expres- 
sions which  represent  the  energy  used  in  heating  the 
conductor  [see  equation  (4)],  in  creating  the  magnetic  field 
around  the  conductor  [see  equation  (6)],  and  in  charging 
the  condenser  [see  equation  (53)]. 
The  equation  of  energy  is 

,  .  idt  i  idt 

•"    lJt    7"   ~~c      * 

The  first  member  of  this  differentia]  equation  eidt 
represents  the  total  energy  supplied  to  the  circuit  in  the 
time  d  t.  A  part  of  this  energy  represented  by  Ri* dt  is 

used  in  heating  the  conductor.     A  second  part  L  i  -T-  d  t  is 

expended   in  creating   a  magnetic  field  in  the  space  sur- 
rounding the  conductor.      A  third   part,  represented  by 

t 

—^ ,  is  expended  in  charging  the  condenser.  Equa- 
tion (85)  is  the  general  differential  equation  of  energy,  in 
terms  of  the  current  which  flows  in  the  circuit,  the  E.  M.  F. 
which  drives  the  current,  and  the  time,  for  a  circuit  con- 
taining resistance,  self-induction,  and  capacity  in  series. 

This  equation  of  energy  may  be  expressed  as  a  differen- 
tial equation  in  terms  of  the  quantity  of  electricity  in  the  con- 
denser, that  is,  the  charge  of  the  condenser,  the  E.  M.  F.,  and 

the  time,  by  means  of  the  relation  dq  =  idt,  o?q  =  fidt. 
On  substituting  in  (85)  i  =  -TT,  we  have 


i  d  tfi  d 


(86)     edt  =  Jl 

dt  (dt  )  dt*  dt         '    C  dt 

Each  term  in  this  equation  is  equal  to  the  correspond- 
ing term  in  equation  (85),  since  it  is  obtained  by  direct 


RESISTANCE,   SELF  INDUCTION,   AND   CAPACITY.      83 

substitution.     The  first  member,  e  -TT  d  t,  is  the  total  energy 

supplied  to  the  circuit,  and  the  three  terms  of  the  second 
member  represent  the  three  ways  in  which  this  energy  is 
expended,  viz.,  in  heat,  creating  the  field,  and  charging  the 
condenser. 

If  equation  (85)  is  divided  through  by  idt,  it  becomes 
iin  equation  of  E.  M.  F.'s,  thus  : 

idt 


If  equation  (86)  is  divided  through  by  -77  d  t,  it  likewise 

d  t 

becomes  an  equation  of  E.  M.  F.  's,  thus  : 


These  are  equations  of  E.  M.  F.'s  :  equation  (87)  in 
terms  of  current,  E.  M.  F.,  and  time  ;  and  equation  (88)  in 
terms  of  the  charge  of  the  condenser,  E.  M.  F.,  and  time. 
Each  term  in  (88)  is  equal  to  the  corresponding  term  in 
(87).  The  first  member,  e,  is  the  E.  M.  F.  impressed  upon 
the  circuit.  That  part  of  e  necessary  to  overcome  the 

resistance  is  Hi,  or  R  -jr.     That  part  of  e  necessary  to 

overcome  the  counter  E.  M.  F.  of  self-induction  is  L  -j-A.  or 

df 

L  j-p'     The  third  part  of  e,  necessary  to  overcome  the 

Ct  L 


counter  E.  M.  F.  of  the  condenser,  is  —  —  —  ,  or  ±-. 

C  C 

These  differential  equations  may  be  written  in  forms 
more  convenient  for  solving.     Differentiating  equation  (87) 


84  CIRCUITS  CONTAINING 

with   regard   to   t,  to  free  it  from  the  integral   sign,  we 
obtain 

d*i      fidi        i          lde 


By  transposition  (88)  becomes 

d¥+~L  dT  +  L~C=  L 

We  know  that  the  impressed  E.  M.  F.  has  one  value  at 
one  particular  time  and  is  therefore  a  single-valued  func- 
tion of  the  time,  that  is,  e  =f(t).  When  we  introduce  this 
relation  into  (89)  and  (90),  the  general  solution  of  each  of 
these  equations  may  be  readily  obtained.  The  solution  of 
equation  (89)  will  give  the  value  of  the  current  at  any  time, 
and  the  solution  of  equation  (90)  will  give  the  value  of  the 
charge  of  the  condenser  at  any  time. 

de 
If  e  =f(t\  and  -TT-  =/'  (t),  upon  substitution  in  (89)  and 

(90),  we  have 

d*  i      R  d  i        i 


tfq       Rdq 

df+L  d~t 


GENERAL  SOLUTION  FOR  CURRENT  AT  ANY  TIME. 

In  solving  equation  (91)  to  obtain  the  value  of  the  cur- 
rent at  any  time,  it  is  convenient  to  make  use  of  the  sym- 
bolic method  for  linear  equations.  (See  page  101,  Johnson's 
Differential  Equations.) 


RESISTANCE,  SELF  INDUCTION,   AND  CAPACITY.      85 
Writing  (91)  in  symbolic  form,  we  have 

1 

i  =  ff  m,    or 


<93>  » = 


1     \  j    w 
LO\ 


Resolving  the  inverse  operator,   -       — ~ — 

partial  fractions,  we  have  the  identical  equation 

I  T.  n 

(94)  -wr-       -r-  -  77^^ 


L  LC 

1 


EC  -  Vlt*C*   -4:1,0 


RC+VR*C* 

9  T.  a 
(95)  Let  T,  = 

and  T.  — 


Placing  these  values  in  (94),  and  substituting  (94)  in  (93), 
we  obtain 

(96)     i  =  —  =   —     __ 

2   2        " 


Each  term  of  equation  (96),  equated  separately  to  t, 
forms  a  linear  equation  of  the  first  order.  This  will  be 
evident  when  we  consider  the  linear  equation  of  the  first 


86  CIRCUITS  CONTAINING 

d  y 
order  between  the  variables  x  and  y,  viz.,  -,-  +  ay  =f(x). 

CL  3C 

When  written  in  the  symbolic  form  this  becomes 
(D  +  a)  y  =/(*), 

(97)  or  y  =  -^-a/(x).      _   -•;.   •     .  "|| 

The  solution  of  this  linear  equation  of  the  first  order  is 
known  to  be  (see  Johnson's  Diff.  Equations,  page  31) 

(98)  y  =  e-  axfe  a*f(x)dx  +  c  e~  a  x  . 


Here  c  is  the  arbitrary  constant  of  integration,  and  none 
other  must  be  added  when  the  integration  is  performed. 
By  equating  (97)  and  (98),  we  have 


ax 

e 


J-  n  f    \  -ax    /* 

D  _)-  a/  fo)  =  6       J 
If  we  replace  a,  in  this  general  formula,  by  the  constant 


j,  and/(x)  by/7  (f),  we  have 


But  this  is  the  value  of  the  first  term  in  the  parenthesis 
of  equation  (96).  The  value  of  the  second  term  in  that 
parenthesis  may  be  found  in  a  similar  manner,  and  (96) 
may  finally  be  written 


(99)    i= 


I 
RESISTANCE,   SELF  INDUCTION,   AND   CAPACITY.       87 

This  is  the  general  solution  of  equation  (91)  and  gives 
the  current  which  flows  at  any  time  in  a  circuit  having 
resistance,  self-induction,  and  capacity. 

Since  the  differential  equation  (92)  for  the  charge  be- 
comes identical  with  the  differential  equation  (91)  for  the 
current  when  we  write  /'  (t)  instead  of  /  (t),  and  since  / 
denotes  any  arbitrary  single  -valued  function  whatever,  we 
may  in  the  general  solution  (99)  suppress  the  accents  on 
the  arbitrary  functions  and  write  the  solution  for  q.  Thus, 


<100>   '- 


PARTICULAR   ELECTROMOTIVE   FORCES. 

These  equations,  (99)  and  (100),  express  the  values  of 
the  current  and  charge  at  any  time,  when  the  impressed 
E.  M.  F.  is  anything  whatever,  since  /  is  any  arbitrary 
single-valued  function  whatever. 

There  are  four  cases,  covering  all  possible  ones,  which 
arise  according  to  the  nature  of  the  impressed  E.  M.  F. 
These  are  : 

Case     I.     e=/(Q  =  0. 

Case    II.     e  =.f(t)  =  E  =.  constant. 

Case  III.     e  =  f(t)  =  #sin  GO  t. 


Case  IY.     e  =f(t)  =          E  sin  (b  GO  t  +  6). 


The  meaning  of  the  first  assumption  is  that  the  im- 
pressed E.  M.  F.  is  to  be  zero  at  every  point  of  time.  This 
condition  is  fulfilled  if  we  charge  a  condenser  with  some 


88  CIRCUITS  CONTAINING 

quantity  Q,  and  then  suddenly  remove  the  impressed 
E.  M.  F.,  that  is,  if  we  connect  the  two  plates  of  the  con- 
denser by  a  conductor  so  as  to  discharge  it.  The  im- 
pressed E.  M.  F.  remains  zero  at  every  point  of  time  after 
the  removal  of  the  source  of  E.  M.  P.,  and  consequently 
satisfies  the  condition  e  =f(t)  =  0.  The  solutions  of  the 
differential  equations  under  this  assumption  give  the  cur- 
rent at  any  time  flowing  in  the  circuit,  and  the  charge 
an  any  time  remaining  in  the  condenser,  when  an  im- 
pressed E.  M.  F.  is  suddenly  removed  from  the  circuit. 
It  may  be  any  circuit  whatever  containing  any  combination 
-of  resistance,  self-induction,  and  capacity,  that  is,  a  circuit 
containing  R  and  Z  alone,  R  and  C  alone,  or  ./?,  Z,  and  C 
together.  In  case  the  circuit  has  R  and  C,  or  /?,  Z,  and  (7, 
the  solutions  will  give  the  current  i  and  quantity  q  at  any 
time  during  the  discharge  of  the  condenser.  If  the  circuit 
contains  R  and  Z  alone,  the  solution  will  give  the  current 
at  any  time  as  it  dies  away  after  the  removal  of  the 
E.  M.  F. 

When  we  assume  e  ~f(t)  —  E  —  a  constant,  we  mean 
that  the  E.  M.  F.  is  to  be  equal  to  E  at  every  point  of 
time.  This  condition  will  be  fulfilled  if  the  source  of 
E.  M.  F.  in  any  circuit  is  suddenly  changed  from  one  con- 
stant value  to  another  constant  value,  either  of  which  may 
be  zero.  If  the  circuit  contains  R  and  (7,  or  7?,  Z,  and  Ct 
the  solutions  give  the  current  flowing  in  the  conductor 
and  the  charge  of  the  condenser  at  any  time  after  the 
change  in  the  E.  M.  F.  If  the  circuit  contains  R  and  Z 
only,  the  solution  gives  the  value  of  the  current  at  any  time 
as  it  changes  to  its  final  steady  value. 

The  third  assumption,  e  =  E  sin  GO  t,  means  that  the 
circuit  contains  an  impressed  E.  M.  F.  varying  harmoni- 
cally with  the  time.  The  solutions  of  the  general  equa- 
tions for  q  and  i  show  that  when  the  impressed  E.  M.  F.  is 
harmonic,  both  the  current  and  the  charge  are  likewise 


, 
RESISTANCE,   SELF  INDUCTION,   AND    CAPACITY.       89 

simple  sine-functions  of  the  time,  having  the  same  period 
as  the  E.  M.  F. 


The   fourth   assumption,  e  =  ^>  E  sin  (b  cot-{-  #),—  - 

E,  6,  (9. 

where  b  takes  in  succession  any  integer  values,  —  means  that 
the  circuit  contains  an  impressed  E.  M.  F.  which  is  any 
periodic  function  of  the  time  whatsoever. 

The  solution  and  discussion  of  these  four  cases  will  be 
considered  in  the  following  chapters, 


CHAPTER  VII. 

CIRCUITS  CONTAINING   RESISTANCE,  SELF  INDUCTION, 
AND  CAPACITY. 

CASE  I.    DISCHARGE. 

CONTENTS  : — Integral  and  differential  equations  when  e  =f(t)  =  0.  Sir 
Win.  Thomson's  solution,  i  equation  with  value  of  T  replaced.  Three 
forms  of  i  and  q  equations.  To  transform  the  /-equation  to  a  real  form 
when  R*  G  is  less  than  4  L.  To  derive  the  solutions  from  the  differen- 
tial equations  when  IP  C=  4  L. 

Non-oscillatory  Discharge. 

Determination  of  constants.  Complete  solution.  Value  of  T  re- 
placed. Current  and  charge  curves  for  a  particular  circuit.  Time  of 
maximum  current.  Equation  (125)  applied  to  a  circuit  containing  resist- 
ance and  self-induction  only,  and  to  a  circuit  containing  resistance 
and  capacity  only. 

Oscillatory  Discharge. 

Determination  of  constants.  Complete  solution  for  i  and  q.  Current 
and  charge  curves  for  a  particular  circuit. 

Discharge  of  Condenser  when  R*  G  —  4  L. 

Determination  of  constants  Complete  solutions  for  i  and  q.  Figure 
showing  method  of  constructing  the  current  and  charge  curves.  Curves 
for  i  and  q  in  a  particular  circuit. 

IN  tins  chapter  the  case  will  be  discussed  in  which  the 
impressed  electromotive  force  is  suddenly  removed  from 
the  circuit  or  reduced  to  zero  ;  that  is,  e  =f(t)  =  0.  When 
a  current  has  been  flowing  in  a  circuit  and  the  source  of 
electromotive  force  has  been  suddenly  removed,  the  cur- 
rent continues  to  flow  for  an  appreciable  time  before 

90 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.       91 

dying  entirely  away.  The  value  of  the  current  at  any 
time  may  be  ascertained  by  applying  the  general  equation 
(99)  to  this  particular  case. 

As  another  example,  we  may  have  a  condenser  or 
Leyden  jar  charged  to  a  certain  difference  of  potential,  and 
the  source  of  potential  then  removed.  If  we  now  connect 
the  two  plates  of  the  condenser  or  coatings  of  the  jar  with 
a  conducting  wire,  a  current  flows  through  the  wire  and 
the  condenser  is  discharged.  The  source  of  potential  was 
previously  removed,  and  so  e  =  f(t]  =  0.  The  general 
equations  (99)  and  (100)  can  be  applied  to  this  particular 
case,  enabling  us  to  ascertain  the  current  which  flows  at 
any  time  in  the  circuit  and  the  charge  remaining  in  the 
condenser. 

Since  f(t)  —  0,  the  first  derivative  is  f'(t)  —  0  ;  and  if 
the  value/'  (t)  —  0  is  substituted  in  the  general  equation 
(99)  for  current,  and  the  value  f(t)  =  0  in  equation  (100) 
for  charge,  we  have 


(101)  i  =  Cle   Tl  +  c,e  T\ 


(102)  q=c,e      '  +  cte     \ 

Had  the  value  e  =  0  =f(t)  been  substituted  in  the 
differential  equation  (92),  and  f  (t)  =0  in  equation  (91), 
we  should  have  had 


It   is   to   be   noted   that   the   form  of   the  differential 
equation  for  i  is  identical  with  that  for  q.     Hence  their 


92  CIRCUITS  CONTAINING 

integrals  (101)  and  (102)  have  the  same  form,  although  with 
different  arbitrary  constants  of  integration.  The  solutions 
of  the  differential  equations  '(103)  and  (104) — which  are 
identical  with  (91)  and  (92)  when  their  second  members 
are  zero — give  what  is  called  the  "  complementary  function  " 
(see  Johnson's  Differential  Equations,  Art.  94).  The 
complementary  function  contains  all  the  arbitrary  con- 
stants of  integration.  The  sum  of  the  particular  integral 
— found  to  satisfy  equations  (91)  and  (92)  when  the  second 
member  is  not  zero — and  the  complementary  function 
gives  the  complete  integral  of  the  general  differential 
equations  (91)  or  (92). 

The  particular  case  of  the  discharge  of  a  condenser 
through  a  circuit  possessing  resistance  and  self-induction 
has  been  fully  discussed  by  Sir  Wm.  Thomson  and  was 
published  as  early  as  1853  in  the  Philosophical  Magazine. 
He  obtained  equation  (102)  as  his  result,  which  he  showed 
could  be  expressed  in  two  different  forms,  according  as  T^ 
and  T^  are  real  or  imaginary. 

Writing  equation  (101)  in  full,  by  replacing  the  values 
of  Tl  and  Ta  given  in  (95),  we  have 


(105)        i  =  Cl 


If  the  value  of  IF  C  is  greater  than  4Z,  the  value  of  i  is 
real  ;  but  if  7?2  C  is  less  than  4  Z,  i  apparently  assumes  an 
imaginary  form.  It  will  be  shown,  however,  that  i  can 
by  a  trigonometric  transformation  be  expressed  in  a  real 
form  when  7?2  C  is  less  than  4Z. 

When  72?  C  is  equal  to  4  L  and  we  have  the  critical  case, 
it  is  evident  that  the  two  terms  of  equation  (105)  may  be 
written  as  one,  and  thus  the  two  arbitrary  constants  com- 
bine into  one.  The  complete  solution,  which  must  contain 
two  arbitrary  constants,  inasmuch  as  it  is  derived  from  a 


RESISTANCE.  SELF  INDUCTION,   AND   CAPACITY.       93 

differential  equation  of  the  second  order,  cannot  be  readily 
obtained  in  this  case  from  (105)  ;  but  it  will  be  directly 
obtained  from  the  differential  equations  (103)  and  (104). 

TO   TRANSFORM   EQUATION   (105)   TO   A   REAL   FORM   WHEN 
R'C  IS   LESS    THAN   4  Z. 

_  21 

After  factoring  out  the  common  factor  e  2L,  we  may 
write  (105)  in  another  form,  thus  : 


_  ,  _  -  .  . 

(106)     i=e      L\cte      ™         +  c,e~ 

Here  j  is  used  to  represent   V—I.     If  we  write 


a 

then  (106)  becomes 

_ 

(108)  ^' 

The  sine  and  cosine  may  be  written  in  exponential  form* 
thus  : 


-    € 
(109)     sin  6  =  -   —~  --  ,    and    cos  0  = 


*  By  Maclaurin's  theorem  for  the  expansion  of  a  function  into  a  series, 
the  sine  and  cosine  may  be  developed  into  the  following  series  : 


a, 


Also  the  development  of  e     into  series  gives 


1-2-3-4-5-6      •?l-2-3.4. 5-6-7  +  1-2-3-4-5-6-7-! 


94  CIRCUITS  CONTAINING 

JQ 
Therefore          cos  6  -f-  j  sin  0  =  e   , 

-jQ 
and        cos  0  —  j  sin  6  —  e 

Multiplying  through  by  cl  and  c2,  respectively,  and  add- 
ing, we  have 

J0  -jQ 

(110)   c^e     -f-  cae        =i  (c,  -f-  ca)  cos  #  +  fo  —  ca)y  sin  6. 

If  Cj  and  ca  are  conjugate    imaginary  quantities,  they 
may  be  written 

_AB 


Multiplying  (1)  by  j  and  adding  to  (2),  we  find  that  the  resulting  series  is 
identical  with  (3).     Hence  we  obtain 

i* 

(4)  cos0+jsiu  0  =  e    . 
-JO 

The  expansion  of  e        into  series  gives 

—JO  0*  03  04  Q5 

(5)  e        =  1  -  "  '    ' 


1.2«8.4-5-e^ 


Multiplying  (1)  by  J  and  subtracting  from  (2),  the  resulting  series  is  iden- 
tical with  (5).     Hence  we  have 

-JQ 

(6)  cos  0  —  .;'  sin  0  =  e 

Adding  equations  (4)  and  (6)  and  dividing  by  2,  we  get 

JO        -JO 

(7) 


a 

Subtracting  (6)  from  (4)  and  dividing  by  2j,  we  have 


(8)  sin  0  =  ~ 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.       95 

where  A  and  B  are  both  real  quantities.     Taking  the  sum 
and  difference, 

c,  +  ca  =  A, 


and  substituting  these  values  in  (110),  we  have 


(111)  c,  e    +  c2  e  ~      =  A  cos  0  +  B  sin  0, 

where  cl  and  ca  are  imaginary,  while  A  and  .Z?  are  real 
quantities.     Substituting  (111)  in  (108),  we  obtain 

_Rt 
o  r 

(112)  i  =  e       (AcosG  +  JB  sin  0). 

By  the  trigonometric  formula  [see  Chapter  III,  equa- 
tion (27)], 


A  cos  6Bsm6  =  VA*      B*  sin 


we  may  finally  write  equation  (112),  after   restoring   the 
value  of  0  from  (107),  in  the  form 


_m 
(113)          i  =  Ae    2L 


where  ^4  and  $  are  the  arbitrary  constants  of  integration. 
Here  A  is  not  the  same  as  in  equation  (112),  but  stands  for 

A 

VA*  +  B\  and  #  stands  for  tan'1-^  .      This  equation  is 

the  equivalent  of  (105).     It  is  real  when  (105)  is  imaginary 
and  imaginary  when  (105)  is  real. 


96  CIRCUITS  CONTAINING 

TO     DERIVE    THE    SOLUTIONS    FROM    THE    DIFFERENTIAL    EQUA- 
TIONS WHEN   P?  C  =  4  L. 


If  ICC  is  equal  to  4Z,  then  the  differential  equations 
(103)  and  (104)  become 

d*i      Edi        R*  . 

0'  and 


d*q       Rdq 


Upon  substituting  i  =  em  *  ,  we  have 
(116)  m'  +      m 


which  is  seen  to  be  a  perfect  square  as  it  stands,  and  con- 
sequently   the     two    values     of    m    become    equal,    and 

TT> 

m  —  —  5-7.     When  there  are  equal  roots,  the  solution  is 


of  the  form 

i  =  c,  em  l  -(-  ca  t  em  * 

(see  Johnson's  Diff.  Equations,  page  95)  ;  or,  replacing  m  by 

-p 
its  value,  —    -,  we  have  as  the  complete  solutions 


_m  _m 

(117)  i  =  c,ii'L  +  c,te    2L, 

_R_t  _Rt 

(118)  q  =  c'i2L  +  c"te    2l  . 

Eeturning  to  equation  (103),  we  may  write  its  solution 
(101),  the  complementary  function,  in  three  different  real 
forms,  according  as  the  value  of  7?2  C  is  greater  than,  less 
than,  or  equal  to  4Z.  These  forms  are  : 


RESISTANCE,  SELF  INDUCTION,  AND  CAPACITY.       97 
When  R*  C  >  4  Z, 

_  RC  -VlPCf-4LCt  _  RC  + 

(119)     i  =  Cle  ~  +  cae 

<4Z, 

Rt 


(120)    <= 


_R_t  _R_t 

(121)  i  =  Cle    2f'  +  c,te    2L. 

The  value  of  the  charge  q  given  by  equation  (102),  being 
of  the  same  form  as  (101),  may  take  three  different  forms 
according  as  J2*Cis  greater  than,  less  than,  or  equal  to  4Z  ; 
and  these  forms  only  differ  from  the  above  in  the  arbitrary 
constants,  thus : 


_  RC  -  *Wc*-4LC't  RC-\- 

(122)  q  =  c'e  *LC  "  2LC 
When  R*C  <  4Z, 

(123)  ,.^--si 


(124) 


Bt  _R_t 

2l        "        2L 


The  constants  of  integration  in  these  equations  are  de- 
termined by  the  initial  conditions  imposed  by  the  problem. 
For  instance,  if  a  condenser  charged  with  a  quantity  Q  is 
suddenly  discharged  through  a  circuit  with  resistance  and 
self-induction,  we  may  count  the  time  from  the  moment  of 
discharge,  and  thus  have  q  =  Q  and  i  =  0  when  t  =  0,  and 
q  =  0  and  i  =  0  when  t  =  GO  . 


98  CIRCUITS  CONTAINING 

NON-OSCILLATORY   DISCHARGE. 

Determination  of  Constants. — The  equations  (ll9)  and 
(122)  may  be  written  as  in  (101)  and  (102),  in  terms  of  the 
time- constants  Tl  and  T^  [see  (95)],  thus  : 

_ '  L  -  L 

(125)  trrC.e^+C.6     T\ 

_  t_  _  t_ 

(126)  2  =  c'e~r>+c"e~r«. 

The  arbitrary  constants  c, ,  c2 ,  c',  c"  of  these  equations 
will  be  determined  according  to  the  conditions  mentioned 
above,  viz.,  when  t  =  0,  i  =  0  and  q  =  Q ;  when  t  =  GO  , 
i  =  0  and  q  =  0.  Substituting  in  (125)  t  =  0  when  t  =  0, 
and  in  (126)  q  =  Q  when  t  =  0,  we  have 

0  =  c,  +  ca,     or    c,  =  —  ca. 

(127)  Q  -  c'  +  c". 

Since  we  have  the  relation  d  q  =  i  d  t,  we  may  differen- 
tiate (126)  and  write 


Equating  this  and  (125),  we  find 


,  =  —  -r,     or    c  =  - 


,     or    c    =  -c^ 


Kemembering  that  c,  =  —  c2  ,  we  may  write 

C"  =  CIT;. 

Adding  c'  and  c", 

c'  +  c"  =  cl(T,-  T,)  =  Q.     [See  (127)]. 


RESISTANCE,  SELF  INDUCTION,   AND  CAPACITY.       99 
Hence       c,  =  7== ^; 


ca  = 

T 

Q 
,-r, 

V 

QT> 

T 

.-21,5 

or. 

Substituting  in  (125)  and  (126)  the  constants  c,  ,  ca  ,  c',  c", 
as  finally  determined,  we  have 


<i29) 


Discussion  of  Non  oscillatory  Discharge.  —  These  equa- 
tions give  the  complete  solution  and  express  the  current 
or  the  charge  at  any  time  after  discharge  (see  Fleming's 
"  Alternate  Current  Transformer,"  Yol.  I.  page  376).  They 
show  that  if  we  have  the  relation  R*  G  >  4  Z,  the  dis- 
charge is  a  gradual  dying  away  without  oscillation.  Since 
T1  and  T^  are  each  of  them  positive  when  R*G  >  4Z  [see 
(95)],  i  or  q  may  be  represented  geometrically  as  the 
difference  of  two  decreasing  logarithmic  curves.  To  see 
this  more  clearly,  the  values  of  the  time-constants  Tl  and 
T^  may  be  substituted  in  the  coefficients  of  equations  (128) 
and  (129).  The  result  is 


(130)       i  = 


100  CIRCUITS  CONTAINING 


\ 

2  ( 


These  equations  may  be  more  easily  explained  by  refer- 
ring to  Figs.  20  and  21,  which  represent  the  plot  of  these 
equations  for  particular  assumed  values  of  R,  L,  and  C. 
The  values  assumed  for  the  constants  of  the  circuit  are 

R  =  100  ohms,     L  =  .0016  henrys,     (7=1  microfarad. 

By  calculating  the  values  of  T7,  and  T^  [equation  (95)], 
Tt  =  8  X  10-  5,  and  T2  =  2  X  10  -5,  the  equations  (130)  and 
(131),  with  these  particular  values,  become 


< 


(133)  g= 


2X10-5 


If  the  condenser  was  charged  to  a  potential  of  2000 
volts,  the  capacity  being  .000001  farads,  the  charge  is  .002 
coulombs.  Substituting  this  value  for  Q,  we  have 


t  =  33.33    e     X-    -e 


2X10-5 


where  i  is  in  amperes  and  q  in  coulombs. 

In  Fig.  20,  curves  I.  and  II.  represent  the  two  compo- 
nent logarithmic  curves,   corresponding   to  the  first   and 


RESISTANCE,  SELF  INDUCTION,  AND  CAPACITY.      101 

second  terms,  respectively,  of  equation  (130),whose  difference 
gives  the  resultant  current  curve  III.  Curve  II.,  cor- 
responding to  the  second  term,  has  the  larger  time-constant, 
and  is  therefore  the  more  important  curve.  The  area  in- 


20xlO~5        Seconds      .     30xlOa 


FIG.  20.— CURVE  SHOWING  CURRENT  DURING  NON-OSCILLATORY  DIS- 
CHARGE OF  CONDENSER  WITH  CAPACITY  (7=1  MICROFARAD, 
THROUGH  A  CIRCUIT  WITH  RESISTANCE  J?  =  100  OHMS,  AND  SELF- 
INDUCTION  L  =  .0016  HENRYS,  WHEN  ORIGINALLY  CHARGED  TO  A 
POTENTIAL  OF  2000  VOLTS. 

eluded  between  curve  III.  and  the  axis  of  abscissae  is  equal 
to    Cid  t  =  Q,  and  is  therefore  independent  of  the  constants 

of  the  circuit  through  which  the  condenser  is  discharged. 

The  current  is  a  maximum  at  a  point  which  may  be  de- 
termined by  differentiating  equation  (130)  and  equating  the 
first  derivative  to  zero  in  the  usual  manner  for  a  maximum. 


102 


CIRCUITS  CONTAINING 


The  time  tm  at  which  the  current  is  a  maximum  is  thus 
found  to  be 


FIG.  21. — CURVE  SHOWING  NON-OSCILLATORY  DISCHARGE  OP  A  CON- 
DENSER, WITH  CAPACITY  (7=1  MICROFARAD,  THROUGH  A  CIRCUIT 
WITH  RESISTANCE  R  =  100  OHMS,  AND  SELF-INDUCTION  L  =  .0016 
HENRYS,  WHEN  ORIGINALLY  CHARGED  TO  A  POTENTIAL  OF  2000 
VOLTS. 

Substituting  in  (134)  the  particular  values  used  in  plot- 
ting Fig.  20,  we  find  the  time  when  the  current  is  a  maxi- 
mum to  be 

tm=3.7S  X  10 -5. 

In  Fig.  21  curves  I.  and  II.  are  the  two  component  loga- 
rithmic curves,  corresponding  to  the  first  and  second  terms, 
respectively,  of  equation  (131)  for  charge.  Curve  III.  is 
plotted  by  subtractiug  II.  from  L,  and  represents  the 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.     103 

charge  of  the  condenser  at  any  time.  It  is  noticeable  that 
the  upper  curve,  I.,  has  the  larger  initial  value,  and  as  Tl 
is  larger  than  Tt,  decreases  the  slower.  It  is  therefore 
this  curve  which  is  the  more  important  in  determining  the 
-discharge  of  the  condenser. 

EQUATION  (125)  APPLIES  TO  A  CIECUIT    CONTAINING    RESISTANCE 
AND  SELF-INDUCTION  ONLY. 

If  there  is  no  condenser  in  the  circuit,  as  explained  in 
Chapter  IV.,  it  is  equivalent  to  saying  that  there  is  a  con- 
denser of  infinite  capacity  in  the  circuit.  Substituting 
C  =  oo  in  the  equation  (95)  for  the  time-constants,  we  have 


EC-V  I?C*-±LC 
T 


According  to  equation  (101),  we  have  the  value  of  the 
current  at  any  time 


Substituting  in  this  equation  the  values  of  Tl  and  T9 
above,  we  have 


When  t  =  0,  i  =  /,  that  is,  the  current  flowing  previous 
to  the  removal  of  the  E.  W.  F.     This  gives 

i  =  I  =  cl  -f-  c,  . 

But  when  t  —  oo  ,  i  =  cv  —  0.     Substituting  these  values 

for  the  constants,  we  have 

st 


.a  result  which  is  well  known  [see  equation  (18)]. 


104  CIRCUITS  CONTAINING 

EQUATION  (125)  APPLIES   TO  A  CIKCUIT  CONTAINING    EESISTANCE 
AND   CAPACITY    ONLY. 

Upon  substituting  L  =  0  in  the  values  of  the  time-con- 
stants T^  and  T^  (95),  the  expressions  become  indeterminate, 
but  can  readily  be  evaluated  by  differentiating  numerator 
and  denominator,  and  then  substituting  L  =  0  as  in  ordi- 
nary vanishing  fractions. 

T 

'  RC~ 

Differentiating  numerator  and  denominator  with  respect 
to  Z,  we  have 


_ 


Now  letting  L  =  0,  we  have 

T,  =  R  0.         Similarly,    T,  =  -  R  C. 

Substituting  these  values  in  equations  (101)  and  (102), 

we  have 

__L  +JL 

(135)  1=0,6      G  +cte     *°. 

(136)  g=cte-**+cte+**.  \ 

c,  and  c4  must  each  be  zero,  or  else  when  t  =  oo  we  would 
have  i  —  oo  and  q  =  oo  .  When  t  =  0,  q  =  Q  =  c3 .  By 
differentiating  (136)  and  equating  to  (135),  we  have 


—  c  e    RC 

— —  O.  c  . 


d£  ~         ^6Y 


cs  Q 

and,  therefore,     •  c,  =  —  -       =  —  -- 


RESISTANCE,  SELF  INDUCTION,  AND   CAPACITY.     105 

Substituting  in  (135)  and  (136)  the  values  found  for  the 
constants  cx  ,  ca  ,  c,  ,  c4  ,  we  have 


(137)  i  =  _  __  e    ™  = 


__ 


(138) 


t 

EC 


These  are  the  well-known  results  for  the  case  of  discharge 
through  a  circuit  with  no  self-induction  [see  equations  (67) 
and  (68)]. 

OSCILLATOEY  DISCHARGE. 

Determination  of  Constants. — In  the  case  of  oscillatory 
discharge,  the  equations  for  current  and  charge  at  any 
time  are 

n  A 

/-mm         •  A    ~2L    - 

(120)      i  =  A  e     L  sin 


Et 


'+«-}• 


The  arbitrary  constants  A,  A',  $,  and  $'  will  be  deter- 
mined according  to  the  same  conditions  as  those  mentioned 
above,  viz.,  when  t  =  0,  i  =  0  and  q  =.  Q  ;  also  when  £  =  oo  , 
i  =  Q  and  q  =  0.  Substituting  in  (120)  i  =  0  when  t  =  0, 
and  in  (123)  q  —  Q  when  t  —  0,  we  have 

0  =  A  sin  $, 

(139)  and         Q  =  A'  sin  <£'. 

Since  A  and  sin  $  are  constants,  and  their  product  is 
zero,  one  of  them  must  be  zero.  But  if  A  is  zero,  i  is  zero 
for  every  point  of  time,  which  is  impossible.  Therefore 

(140)  $  =  0. 


106  CIRCUITS  CONTAINING 

Differentiating  (123)  and  remembering  that  i  =  -77  ,  we 

Ci  t 

have 


R_t 

— sin 


i 


A'e 


Substituting  i  =  0  when  t  =  0, 


- 

0  =  —  R  sm  $'  H  --  7=  --  cos 


4/2    7  /> 

(142)      Hence    *'  =  tan"'- 


By  (139),  ^'=5-     And,  by  (142), 

^  _  Q 


At 


To  determine  the  constant  A,  transform  (141)  by  the 
formula  (27)  so  as  to  write  it  in  terms  of  a  sine  only.  The 
coefficient  of  the  sine  in  the  equation  as  transformed  will  be 


A 


Since  equations  (141)  and  (120)  are  each  equations  for  i, 
we  may  equate  the  coefficients  of  the  sine,  and  have 

A' 

-  ' 

And,  by  (143), 
(144)  A  = 


I 

RESISTANCE,  SELF  INDUCTION,  AND  CAPACITY.     107 

Substituting  A  and  #,  as  determined  in  (144)  and  (140), 
in  equation  (120),  and  substituting  A'  and  $',  as  determined 
in  (143)  and  (142),  in  equation  (123),  we  have 


(145)    i 

2G 

filTI   <    -  •    - 

4Z  C—1FC* 

\j-^vj 
ruffi    n  - 

V±LC  -E*C* 
2  Q  VTC 

£i 

"  2i 

2LC 

Discussion  of  Oscillatory  Discharge. — These  equations 
may  be  more  readily  understood  by  referring  to  Fig.  22, 
in  which  curves  showing  the  current  and  charge,  according 
to  these  equations,  are  drawn  for  the  discharge  of  a  con- 
denser for  particular  values  of  7?,  Z,  and  C,  assumed.  The 
particular  constants  assumed  are 

R  —  100  ohms,   L  =  .0125  henrys,    C  =\  microfarad. 

If  the  condenser  be  originally  charged  to  2000  volts,  Q  = 
.002  coulombs.  On  substituting  these  values  in  (145)  and 
(146),  the  equations  for  current  and  charge  become 

i  =  20e~    '00 'sin  8000 1,    and 

q  =  .00224  e  ~  4mt  sin  (8000 1  +  tan" '  2), 

where  i  is  in  amperes  and  q  in  coulombs.  Curve  I.,  repre- 
senting the  current,  is  a  sine-curve  with  an  amplitude  de- 
creasing according  to  the  logarithmic  curve  20  e 

The  period  is  QAAA  =  .000785  seconds ;    that  is,  there  are 
ouOO 

1275  complete  oscillations  per  second.  A  very  few  oscil- 
lations are  sufficient  for  a  complete  discharge. 

The  charge  at  any  time  is  shown  in  curve  II.,  which  is 
likewise  a  sine-curve  with  an  amplitude  decreasing  accord- 


108 


CIRCUITS  CONTAINING 


-mot 


ing  to  the  logarithmic  curve,  in  this  case  .00224  e 
The  scale  in  Fig.  22  is  such  that  the  same  logarithmic  curve 
is  an  envelope  for  the  current  and  the  charge  curve.  The 
periods  of  the  two  are  the  same,  but  the  curves  differ  in 


FIG.  22.  —  OSCILLATORY  DISCHARGE  OF  A  CONDENSER  WITH  CAPACITY 
C  =  I  MICROFARAD,  THROUGH  A  CIRCUIT  WITH  RESISTANCE  R  — 
100  OHMS,  AND  SELF-INDUCTION  =  L  .0125  HENRYS,  WHEN  ORIGI- 

NALLY   CHARGED    TO    A    POTENTIAL    OF    2000    VOLTS. 

phase  by  an  angle  —  tan"1  2,  that  is,  the  charge  is  ahead  of 
the  current  by  an  angle  of  advance  of  63°  27'. 


DISCHARGE   OF   THE   CONDENSER   WHEN   fi*  C  =  4  L. 

Determination  of  Constants.  —  This  is  the  critical  case  when 
the  discharge  is  just  non-oscillatory.  The  equations  for 
the  current  and  charge,  as  previously  determined,  are 


(121) 
(124) 


-  sir    x; 


Rt 
2L 


+  c"  t  e 


2L 


Rt 
2L 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.     109 

The  arbitrary  constants  of  integration.  c,,  c2,  c',  c",  of  these 
equations  will  be  determined  by  the  same  conditions  as  in 
the  previous  cases,  namely,  when  t  =  0,  i  =  0  and  q  =  Q. 
Equations  (121)  and  (124)  then  become 

(147)  0  =  ct, 


Differentiating  equation  (124)  and  substituting  Q  for  c',  we 
obtain 


But  when  t  =  0,  i  =  0  ;  therefore 


Equating  equations  (121)  and  (148)  and  replacing  the  values 
for  the  constants  given  in  (147)  and  (149),  we  have 

(150)          -'  C'  =  -1TZ7- 

Substituting  for  Q  its  value  E  C,  and  for  J?2  its  equivalent, 

in  this  particular  case  —  T  ,  (150)  becomes 

o 

E 


Having  thus  determined  the  values  of  the  arbitrary  con- 
stants, the  equations  for  current  and  charge  may  be  written 

F     --- 
(151)  i=-*teZL. 


(152)  '"    '   "    ~:m-    2L 

Discussion  of  Discharge  when  B*  C  —  4  L. — These  equa- 
tions give  the  value  of  the  current  and  charge  at  any 
time  during  the  discharge  of  the  condenser  in  the  case 


110 


CIRCUITS  CONTAINING 


where  the  discharge  is  just  non-oscillatory.  This  case 
is  often  called  the  case  of  quickest  discharge,  as  was  pointed 
out  by  Dr.  W.  E.  Sumpner  in  the  Philosophical  Magazine, 
and  afterwards  discussed  by  Dr.  Oliver  Lodge  in  the  Elec- 
trician for  May  18,  1888. 

The  curve  representing  the  plot  of  the  equation  (151) 
for  the  current  may  be  drawn  as  indicated  in  Fig.  23.     A 


2- 


Time 


FIG.  23. — SHOWING  METHOD  OF  CONSTRUCTING  THE  CURRENT  CURVE 
IN  THE  CASE  WHERE  .R2  C  =  4  L. 

-ri 

logarithmic  curve  L,  having  its  initial  value  y-  and  time- 

2  L 
constant  —&  ,  is  drawn  to  represent  the  equation  as  it  would 

be  with  t  omitted  from  the  coefficient ;  and  each  ordinate 
is  then  multiplied  by  the  ordinate  of  a  straight  line  II., 
which  passes  through  the  origin  and  represents  the  uniform 
increase  of  the  time  t.  The  product  of  the  ordinates  of 
curves  I.  and  II.  at  each  point  gives  the  ordinate  of  the 
current  curve  III.  at  that  point.  When  actual  values  of  R, 
L,  and  G  are  assumed,  it  is  found  to  be  difficult  to  repre- 
sent these  curves  to  scale,  so  that  Fig.  23  is  shown  simply 
as  an  illustration  of  the  method  of  constructing  the  current 
curve.  Curve  L,  Fig.  25,  represents  the  current  in  an 
actual  case  where  R  =  100  ohms,  L  —  2.5  henry s,  and  C  = 
1000  microfarads,  the  condenser  being  originally  charged 
to  a  potential  of  2000  volts. 

The   method   of   constructing   the   curve,  showing   the 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.     Ill 


charge  left  in  the  condenser  at  any  time,  is  given  in  Fig.  24, 
and  is  similar  to  the  method  just  shown  for  constructing 


FIG.  24. — SHOWING  THE  METHOD  OF  CONSTRUCTING  THE  CURVE  REP- 
RESENTING THE  CHARGE  LEFT  IN  THE  CONDENSER  AT  ANY  TIME 
AFTER  DISCHARGE. 

the  current  curve.  The  difference  is  that  the  straight  line 
passes  through  a  point  one  unit  above  the  origin,  on  the 
vertical  axis,  instead  of  through  the  origin  as  before. 


FIG.  25.— JUST  NON-OSCILLATORY  DISCHARGE  OF  A  CONDENSER  WITH 
CAPACITY  C  =  1000  MICROFARADS,  THROUGH  A  CIRCUIT  WITH 
RESISTANCE  E  =  100  OHMS,  AND  SELF-INDUCTION  L  —  2.5  HENRYS. 

The  logarithmic  curve  has  the  initial  value  Q  and  a  time- 

2Z 

constant  -~- .     The  curve  showing  the  charge  for  the  actual 

case  where  7?  =  100  ohms,  L  =  2.5  henrys,  and  C—  1000 
microfarads,  the  condenser  being  originally  charged  to  a 
potential  of  2000  volts,  is  represented  by  curve  II., 

Fig.  25. 


CHAPTER  VIII. 

CIRCUITS  CONTAINING  RESISTANCE,  SELF  INDUCTION,  AND 

CAPACITY. 

CASE  II.    CHARGE. 

CONTENTS:—  Differential  equations  with  e  =  f(t)  =  E.  Solution  of  these 
equations.  Solution  from  the  general  integral  equation..  Three  forms 
of  i  and  q  equations. 

Non  oscillatory  Charging. 

Determination  of  constants.  Complete  solutions  for  *  and  g  with 
constants  determined.  Curves  for  i  and  q  in  a  particular  circuit.  Equa- 
tion (101)  applied  to  a  circuit  containing  resistance  and  self-induction 
only;  also  to  a  circuit  containing  resistance  and  capacity  only. 

Oscillatory  Charging. 

Determination  of  constants  Complete  solutions  £or  i  and  q  with 
constants  determined.  Curves  for  i  and  q  in  a  particular  circuit. 


Charge  of  the  Condenser  when  R^C—4.L. 

Determination  of  constants.     Complete  solutions  for  i  and  q  with 
constants  determined.     Curves  for  i  and  q  in  a  particular  circuit. 

THE  E.  M.  F.,  instead  of  being  zero,  as  in  Case  I.,  is 
assumed  to  be  a  constant  E,  and  e  =f(t)  —  E.  This  is  the 
case  when  an  E.  M.  F.  is  suddenly  changed  from  one  con- 
stant value  to  another  constant  value  in  a  circuit,  and  it 
includes  Case  I.  as  a  particular  case,  since  E  may  be  zero. 
Since  e  is  a  constant,  the  first  derivative  of  e  is  zero,  and, 

112 


RESISTANCE,  SELF  INDUCTION,  AND   CAPACITY.    113 

therefore,/7  (t)  =  0.     Substituting  these  values  of/(£)  and 
'(£)  in  the  differential  equations  (89)  and  (90),  they  become 

d*i      E  di         i 

dT  +  Ldt  +£-&  =  *' 


HKA\  *  Rdq   }      q  E 

(154)       and        -    +  r-Jr  +  j^   =  _. 


It  is  seen  that  the  equation  for  current  (153)  is  identical 
with  that  of  the  previous  case  (103),  while  the  equation  for 

ff 

charge  (154)  has  its  second  member  equal  to  -^,  a  constant, 

jL 

instead  of  being  zero  as  in  equation  (104).  By  substituting 
a  new  variable,  q'  =  q  —  E  6r,  this  equation  may  be  trans- 
formed into  one  having  its  second  member  zero,  thus  : 


The  solutions  of  (153)  and  (155)  are,  as  in  the  previous  case, 

_  L  _  L 

(101)  i  =  c,e    Tl  +  c,e    T\ 

_  L  _  1. 

q>  —  c  e    Tl  -4-  c4  e    T\ 

Eeplacing  the  value  of  q',  and  remembering  Q  =  E  C  = 
the  final  charge,  we  may  write 


(156) 


These  equations,  (101)  and  (156),  for  current  and  charge 
might  have  been  obtained  directly  from  the  general  solu- 
tions (99)  and  (100)  by  substituting/^)  =  E,  and/7  (t)  =  0. 
Upon  substituting  f'(t)  =  0  in  (99),  we  obtain  (101)  directly, 
and  upon  substituting/^)  =  ^in  (100),  we  have 

EC 


CIRCUITS  CONTAINING 
But,  by   the   values   of    T,  and  T,  in  (95),  we  find   that 


-  ±LC\  and  hence  this  equation  is 
identical  with  (156),  as  Q  =  EC. 

As  in  Case  I,  where/(£)  =  0,  the  equations  just  obtained 
for  current  (101)  and  charge  (156),  when  /  (t)  =  E,  assume 
three  forms. 


(157)  i  =  Cle    r 

(158)  g  =  Q  +  c'e    T*+c"e 


-  L 
T*"~T 


(159)       ;  = 


Rt 


(160)       ,  =  '  # 


_R_t  _Rt 

(161)  i  =  c,e    ^L  +  c,te~^L. 

_Rt  _  Rt 

(162)  g=  Q  +  c'e    2L+cffte    *L. 

The  constants  of  integration  must  be  determined  by  the 
conditions  of  the  problem  as  to  the  previous  state  of  the 
circuit,  the  changes  made,  and  the  final  state. 

NON-OSCILLATORY   CHARGING. 

Determination  of  Constants. — The  constants  c, ,  c2 ,  c',  and 
c"  of  equations  (157)  and  ^158)  will  be  determined  by  the 
following  conditions : 

When  t  =  0,       i  =  0     and     q  =  Q0 . 
When  t  =  GO  ,     i  =  0     and     q  =  Q. 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY     115 

This  means  that  the  condenser  is  suddenly  charged  or  dis- 
charged from  the  initial  charge  Q0  to  the  final  charge  Q. 
Determining  the  constants  by  the  same  method  as  in  Case 
I.,  we  find  that 


,  -  T; 


C    = 


r,,_(Q.-Q) 

G  •  rrj  rr 


Substituting  in  (157)  and  (158)  the  values  of  the  constants 
just  determined,  we  have 


(163) 


(164) 


For  Q0  ,  the  original  charge,  we  may  write  CE0  ,  and  for  Q 
the  final  charge,  we  may  write  C  E. 

These  equations  give  the  value  of  the  current  and 
charge  at  any  time  after  the  change  of  E.  M.  F.  from  E0  to 
E  in  a  circuit  with  IF  C  >  4  L.  As  the  equations  now 
stand  in  their  general  form,  they  hold  true  for  either  total 
or  partial  charge  or  discharge  according  to  the  values  of 
EQ  and  E,  and  consequently  QQ  and  $,  assumed.  If  the 
final  charge  is  Q  =  0,  we  have  the  case  of  complete  dis- 
charge and  the  equations  take  the  form  of  (128)  and  (129). 
If  the  original  charge  Q0  =  0,  we  have  the  case  of  charge 
from  zero  to  Q. 


116 


CIRCUITS  CONTAINING 


Disciission  of  Non-oscillatory  Charge. — These  equations 
will  perhaps  be  better  understood  by  referring  to  Fig. 
26,  which  represents  the  equations  with  particular  values 


FIG.  26.— NON-OSCILLATORY  DISCHARGE  OF  A  CONDENSER  WITH  CA- 
PACITY C  =  1  MICROFARAD,  THROUGH  A  CIRCUIT  WITH  RESISTANCE 
R  =  100  OHMS  AND  SELF-INDUCTION  L  =  .0125  HENRYS  WHEN  SUB- 
JECTED TO  A  POTENTIAL  OF  2000  VOLTS. 

assumed.  These  values  are  the  same  as  in  the  pre- 
ceding case,  namely,  R  =  100  ohms,  C  =  1  microfarad, 
L  =  .0016  henrys.  The  condenser  originally  had  no  charge, 
and  when  charged  to  a  potential  of  2000  volts,  has  a 
charge  of  .002  coulombs.  The  current  curve  L,  Fig.  26, 
is  identical  with  curve  III.,  Fig.  20,  which  represents  the 
current  during  discharge.  Curve  II.  representing  the 
charge  is  the  same  as  curve  III.,  Fig.  21,  inverted  and 
plotted  downwards  from  the  horizontal  line  Q  =  .002.  It 
is  noticeable  that  the  ordinates  of  curve  I.,  expressing  the 
current,  are  proportional  to  the  tangents  of  the  angle  of 
inclination  of  curve  II.  at  every  point,  since  the  current 

i=  -=-,  and  -^-  is  the  tangent  of  the  angle  of  inclination  of 

the  curve  of  charge  II.  It  is  seen  that  the  point  of  inflec- 
tion on  curve  II.  comes  at  the  maximum  value  of  the  current 
curve  I.,  as  the  tangent  is  a  maximum  at  this  point.  In- 
deed, curve  I.  might  be  constructed  geometrically  simply 
from  the  foregoing  consideration. 


RESISTANCE,  SELF  INDUCTION,   AND  CAPACITY.     117 

EQUATION  (101)  APPLIES  TO  A   CIRCUIT  CONTAINING   RESISTANCE 

AND  SELF  INDUCTION  ONLY,  IN  THE  CASE  OF  THE  ESTABLISH- 

MENT OF  A  CURRENT  UPON  INSERTING  AN  E.  M.  F. 

In  this  case  there  is  no  condenser  in  the  circuit,  that  is, 
the  capacity  is  infinite.     Substituting  C  =  oo  in  the  values 

of  the  time-constants  (95),  we  have  T^  =  oo  ,  T3  =  -=,  as  in 

-/i 

Case  I.,  where  the  current  dies  away  after  the  removal  of 
the  E.  M.  F.     Substituting  these  values  in  (101),  we  have 


When  t  =  0,      t  =  0  =  c^  +  c,. 

When  t=  oo,    i  —  I=cl.     .•.  ca  =  —  /. 

Substituting  these  values  for  the  constants  c,  and  ca,  we 
have 


«  =  7(1 -.-'•) 


/  is  the  final  steady  value  of  the  current,  and  is  equal  to 

-5;  hence 
H 


(21) 


which  is  the  well-known  expression  for  the  establishment 
of  a  current  in  a  circuit  with  self-induction  [see  equation 
(21),  Chap.  III.]. 

EQUATION  (156)   APPLIES  TO  A  CIRCUIT   CONTAINING  RESISTANCE 
AND  CAPACITY  ONLY  IN  THE  CASE  OF  CHARGING  A  CONDENSER. 

Upon  substituting  L  =  0  in  the  values  of  the  time-con- 
stants I7,  and  T^,  the  expressions  become  indeterminate,  but 


118  CIRCUITS  CONTAINING 

can  be  evaluated  as  before  by  differentiation  of  numerator 
and  denominator  before  substituting  L  =  0.  We  thus  find 
the  values,  when  L  —  0, 


Substituting  these  values  in  the  equations  of  current  (101) 
and  charge  (156),  we  have 

.JL  _J_ 

(165)  i  =  cl€    RC  +  c,e    RC. 


(166)        ;     .         g=Q  +  C3e~^  +  c.e^.  _ 

Q0  is  the  previous  charge  of  the  condenser,  and  Q  the  final 
charge.  The  constants  c2,  c4  must  be  zero,  or  else  when 
t  =  co  we  would  have  i  =  GO  ,  q  —  oo  .  When  t  =  0,  equa- 
tion (166)  becomes 

Q.  =  Q  +  e,.      .:  e.=  Q.~  Q. 

By  differentiating  (166)  and  equating  to  (165),  we  have 


_dq  _          c, 

~ 


RC 


Therefore 


Substituting  in  (165)  and  -(166)  the  values  for  the  constants 
cu  C2J  C3>  C4>  as  determined, 


(167) 


(168)  q=Q  +  (Q0—  Q)e    RC. 

These  equations  are  true  for  the  charge  or  discharge 
from  Q0  to  Q,  through  a  resistance  with  no  self-induction. 
When  the  final  charge  Q  is  zero,  we  have  the  case  of  com- 
plete discharge,  and  the  equations  become  the  same  as  (137) 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.      119 

and  (138).  When  the  original  charge  Q0  is  zero,  we  have 
the  case  of  charging  from  zero  to  Q,  and  equations  (167) 
and  (168)  become 


These  equations  are  identical  with  (72)  and  (71),  already 
obtained  in  Chap.  V.  It  is  noticeable  that  the  current 
equation  is  the  same  as  that  for  discharge  equation  (137), 
and  that  the  charge  equation  is  analogous  to  that  in  the 
case  of  the  establishment  of  the  current  in  a  circuit  with 
resistance  and  self-induction,  equation  (21). 

OSCILLATORY  CHARGING. 

Determination  of  Constants.  —  The  constants  A,  A',  $, 
and  $'  in  equations  (159)  and  (160)  will  be  determined  by 
the  same  conditions  as  before,  namely, 

When  t  =  0,     i  =  0     and     q  =  Q0. 
When  t  =  GO  ,  i  =  0     and  '   g  =  Q. 

The  meaning  of  this  supposition  is  the  same  as  in  the  pre- 
ceding case,  namely,  that  the  condenser  is  suddenly  charged 
or  discharged  from  the  initial  charge  Q0  to  the  final  charge 
Q.  The  constants,  determined  by  the  same  method  as  in 

Case  I.,  are 

2(0.  -0       . 


A= 


120  CIRCUITS  CONTAINING 

With  the  constants  thus   determined,  equations   (159) 
and  (160)  become 


7?  / 


2Z  6y 


V±LC-R*C° 
sm 


We  may  write  CE9  for  the  original  charge  Q9,  and  CE  for 
the  final  charge  $. 

Discussion  of  Oscillatory  Charge. — These  equations  give 
the  value  of  the  current  and  charge  at  any  time  after 
the  change  of  the  electromotive  force  from  E0  to  E  in 
a  circuit  with  M*C9  <  4  L.  As  the  equations  now  stand 
in  their  general  form,  they  are  true  for  either  total  or 
partial  charge  or  discharge,  according  to  the  values 
assigned  to  Q0  and  Q.  If  the  final  charge  Q  is  zero,  we 
have  the  case  of  complete  discharge,  and  the  equations  take 
the  form  of  (145)  and  (146).  When  Q  is  less  than  Q0,  we 
have  partial  discharge  ;  if  Q  is  greater  than  Q0,  we  have 
partial  charging.  If  the  original  charge  Q0  =  0,  we  have 
the  case  of  charge  from  zero  to  Q. 

Fig.  27  illustrates  the  case  of  oscillatory  charge  through 
a  circuit  having  the  same  constants  as  those  of  Fig.  22. 
The  current  curve  I.  is  the  same  as  that  in  Fig.  22,  and  the 
charge,  represented  by  curve  II.,  is  the  same  as  in  that 
figure,  but  inverted  and  plotted  from  the  horizontal  line 
Q  =  .002.  It  is  seen  that  in  charging  the  condenser,  the 
charge  rises  at  first  higher  than  its  final  value,  and  then 
oscillates  about  that  final  value  until  it  has  become  steady. 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.      121 
CHARGE   OF   THE   CONDENSER  WHEN   R*  C  =  4  L. 

Determination  of  Constants. — This  is  the  critical  case, 
where  the  charging  is  just  non-oscillatory.  The  equations 
for  current  and  charge  are 


(161) 


(162) 


q= 


Rt 
2L 


ce 


Rt 


Rt 
2L 


The  initial  charge  is  Q0,  and  the  final  charge  Q.     To 
determine  the  arbitrary  constants  of  integration,  let  t  =  0. 


10~4  Seconds 


FIG.  27.— OSCILLATORY  CHARGE  OP  A  CONDENSER  WITH  CAPACITY  C  = 
1  MICROFARAD,  THROUGH  A  CIRCUIT  WITH  RESISTANCE  R  —  100 
OHMS  AND  SELF-INDUCTION  L  =  .0125  HENRYS  WHEN  SUBJECTED 
TO  A  POTENTIAL  OF  2000  VOLTS. 

Then  i  -  0,  and  q  =  Q0.     Equations  (161)  and  (162)  then 
become 

c,  =  0. 


122  CIRCUITS  CONTAINING 

Differentiating  equation  (162)  and  substituting  the  value  of 
c',  we  have 

7?  / 

/17D  i-**-l(Q*=®*  ,  c-    *£ 

2?-*  2Z  2Z 

When  t  =  0,  i  =  0  ;  therefore 


_ 


Equating  equations  (161)  and  (171),  and  replacing  the  values 
for  Cj,  c',  and  c",  we  have 


4Z! 


If  ^  and  E  are  the  initial  and  final  potentials,  respectively, 
we  may  write  E0C ^for  $0,  and  EC  for  $.  Making  this  sub- 
stitution and  remembering  that  in  this  particular  case 


4Z         . 
=  -77- ,  we  have 


E  -K 


c>  =  -~z 


Eeplacing  the  values  of  the  arbitrary  constants,  the  equa- 
tions (161)  and  (162)  for  current  and  charge  may  be  written 

F_  F      _*« 
(172)  i  =  -±—^te    : 

/      ///\  -B* 

(173) 


Discussion  of  Charge  when  J?2  (7  =  4  Z. — The  current 
curve  in  the  case  of  charging  a  condenser,  represented 
by  equation  (172),  is  the  same  as  in  the  case  of  discharge, 
equation  (151).  It  is  represented  in  Fig.  28,  curve  L, 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY. 

and  may  be  constructed  by  the  method  shown  in  Fig.  23. 
Curve  II.,  Fig.  28,  showing  the  charge  is  constructed  in 


FIG.  28.— JUST  NON-OSCILLATORY  CHARGE  OF  A  CONDENSER  WITH 
CAPACITY  (7.=  1000  MICROFARADS  THROUGH  A  CJRCUIT  WITH 
RESISTANCE  R  =  100  OHMS,  AND  SELF-INDUCTION  L  —  2.5  HENRYS. 

a  similar  manner  to  curve  II.,  Fig.  25,  and,  indeed,  curve 
II.  of  Fig.  28  is  identical  with  curve  II.  of  Fig.  25,  it 
being  inverted  and  plotted  downwards  from  the  horizontal 
line. 


CHAPTER  IX. 

CIRCUITS  CONTAINING  RESISTANCE,   SELF-INDUCTION, 
AND  CAPACITY. 

CASE  III.    SOLUTION  AND  DISCUSSION  FOR  HAKMONIC  E.  M.  F. 

CONTENTS  :— To  find  from  the  general  solutions  the  particular  equations  in 
the  case  of  an  harmonic  E.  M.  F.  Complete  solutions  for  i  and  q. 
These  same  solutions  obtained  directly  from  the  differential  equations. 

Discussion  of  Case  III.     Harmonic  E.  M.  F. 

The  impediment.  Case  A.  Circuits  containing  resistance  and  self- 
induction  only.  Case  B.  Circuits  containing  resistance  and  capacity 
only.  Case  C.  Circuits  containing  resistance  only.  Case  D.  Circuits 
containing  capacity  only. 

Effects  of  Varying  the  Constants  of  a  Circuit. 

First.  Electromotive  force  varied.  Second.  Resistance  varied. 
Third.  Coefficient  of  self-induction  varied.  Fourth.  Capacity  varied. 
Fifth.  The  frequency  varied. 

The  energy  expended  per  second  upon  a  circuit  in  which  an  har- 
monic circuit  is  flowing 

THE  EQUATIONS  FOE  AN  HARMONIC  E.  M.  F.  OBTAINED  FROM 
THE  GENERAL  SOLUTION. 

In  the  preceding  cases  considered,  those  of  discharge  and 
charge,  the  solutions  for  the  value  of  the  current  and  charge 
at  any  time  were  obtained  in  two  ways,  first  from  the  general 
solution,  and  then  directly  from  the  differential  equations,  by 
substituting  e  =f(t)  =  0,  and  e  =  f(t)  —  E,  respectively. 

.124 


RESISTANCE,  SELF  INDUCTION,  AND   CAPACITY.    125 

The  case  of  a  circuit  containing  resistance,  self-induc- 
tion, and  capacity,  in  which  there  is  an  impressed  E.  M.  F. 
varying  harmonically,  will  now  be  considered,  and  the  solu- 
tion derived  first  from  the  general  equations  (99)  and  (100), 
and  then  directly  from  the  differential  equations  (89)  and 
(90).  In  this  case, 

(174)  e=f(t)    =^sinctf£, 

(175)  and         ~  =/'  (t)  =  EGO  cos  cot. 
Substituting  these  values  in  (99)  and  (100),  we  have 


(176)   j= 


-          +  x  _  - 

—  e    T'Je    T*  sin  cotdt  [  +  C3  e    TI  +  C4  e    T 


The  solution  for  q  being  similar  to  that  for  -i,  we  will 
give  the  integration  and  reduction  of  (176)  alone,  and  simply 
give  the  resulting  expression  for  q.  The  integrals  may  be 
found  by  the  formulae  of  reduction  [see  equations  (24)  and 
(25),  Chapter  III.],  obtained  by  integrating  by  parts.  The 
integration  of  each  term  in  (176)  is 


(178) 
ji 

~~  T 


Je    T  cos  GDtdt  =  -j—  1  m  cos  cot -\-cosm  cot  ^  . 

-^r,  -4-   Cs9     I 


126  CIRCUITS  CONTAINING 

For   convenience   in    transformation    and    reduction,   put 

r1=-^r  and  r^—  -=-.     After  making  these  substitutions  in 

•*  I  •*  3 

equation  (176),  we  have 


We  may  simplify  (179)  by  substituting  the  values  of  rl  and 
TI  [see  (95)]. 


"  T,-  2LC 

Then,  after  a  few  simple  algebraic  transformations  in  the 
coefficients  of  the  sine  and  cosine,  (179)  becomes 

(180)  u  —  **•*        •    - 


_L  _L 

T  T 

+  —        —r-        -^coscat  +  c.e       +c,e      . 


This  may  be  transformed  into  a  more  convenient  form  by 
means  of  the  trigonometrical  formula  [see  equation  (27), 
Chapter  III.] 


A  sin  x  +  B  cos  x  —  VA*  +  J52  sin   aJ  +  tair1       , 


RESISTANCE,  SELF  INDUCTION,  AND.  CAPACITY      127 
and  when  transformed  is  written 

(181)  *'=' 


This  is  the  complete  solution  for  the  current  in  a  circuit 
with  resistance,  self-induction,  and  capacity  when  the 
E.  M.  F.  is  harmonic  and  equal  to  E  sin  cot.  The  discussion 
of  this  equation  is  deferred  to  the  latter  part  of  the  chapter. 

To  FIND  THE  EQUATION  FOR  CHARGE. 
The  corresponding  equation  for  charge,  being  the  inte- 
gral of  the  current  according  to  the  relation  q  =    f  id  t,  may 
be  written 

Tjl 

(182)    q  =  - 


Cal 


This  equation  is  the  complete  solution  for  the  charge  in  a 
circuit  with  resistance,  self-induction,  and  capacity,  when 
there  is  an  harmonic  impressed  E.  M.  F. 

To  OBTAIN  THE  SOLUTION  DIRECTLY    FROM  THE  DIFFEREN- 
TIAL EQUATION. 

Let  us  now  proceed  to  obtain  this  same  solution,  equa- 
tion (181),  by  solving  the  original  differential  equation, 
with  the  assumption  that  the  E  M.  F.  varies  harmonically, 


128  CIRCUITS  CONTAINING 

de 
dt 


de 
that  is,e  =  .ZTsin  cot.    Substituting -77  =  E GO  cos  cot  in  the 


differential  equation  (89),  we  have 

d*i      Edi         i        EGO 


This  is  a  linear  equation  of  the  second  order  with  constant 
coefficients.  [See  Johnson's  Differential  Equations,  page 
91].  The  complete  integral  of  such  an  equation  consists  of 
the  sum  of  two  parts,  namely,  the  particular  integral  and 
the  complementary  function.  The  complementary  function 
is  the  integral  obtained  by  equating  the  first  member  to 
zero,  and  contains  two  arbitrary  constants.  The  particular 
integral  contains  no  arbitrary  constants.  The  comple- 
mentary function,  obtained  by  equating  the  first  member  to 
zero  and  solving,  is 

__*.  _  L 

(101)  1  =  0,6    r'+c2e    T\ 

To  find  the  particular  integral,  it  is  convenient  to  use  the 
symbolic  notation 


dt 
With  this  notation  (183)  is  written 


EGO 

—j-  cos  opt 

(184)  or  i  = 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.    129 
Next,  to  find  the  value  of  D*,  we  have 
d  cos  cat 


dt 

d*  cos  Got 


=  D  cos  GO  t  =  —  GO  sin  GO  t, 

=  D*  COS  GO  t  =   —  GO*  COS  GO  t. 


dt* 

Therefore  D*  =  —  GO*. 

Substituting  in  (184)  D*  —  —  GO*,  we  have 

EGO 
(185)     i  =  — TTT-        —*—      — r  cos  ca  t 


EGO 

-1 COS  GO  t. 


Multiplying  numerator  and  denominator  of  the  coefficient 
of  cos  Got\)j  R D  —  (-^  —  L  GO*),  we  obtain 


-(~  -Zor'U 


cos 


Substituting  —  GO*  for  D'\  and  separating  into  two  terms, 
—  E  GoR  D  cos  cot-\-  EGO  (—  —  LGO*\  cos  G?£ 

.  \c/  / 


But  Z>  cos  cot  =  —  GO  sin  ca  £.     Hence 

.C/G?  I  ^   ' 

sin 


EGO    -^  --  LGO  \  cos  GO  t 

\<7  / 


i  /i  \2 

~  -  L  GO*)  R*GO*  +  (-±--L  GO*) 


180  CIRCUITS  CONTAINING 

Tliis  is  the  particular  integral,  to  which  must  be  added  the 
complementary  function  (101)  in  order  to  obtain  the  com- 
plete integral.  The  complete  integral  is  thus  found  to  be 


This  solution  for  the  current  obtained  from  the  differential 
equation  (89)  is  seen  to  be  identical  with  (180),  the  result 
obtained  from  the  general  solution  (99).  The  solution  for 
charge  could  be  obtained  in  a  similar  manner  from  the  dif- 
ferential equation  (90). 

DISCUSSION  or  CASE  III. — HARMONIC  E.  M.  F. 

These  solutions,  (181)  and  (182),  show  that,  after  a  very 
short  time  has  elapsed,  so  that  the  exponential  terms  con- 
taining the  arbitrary  constants  of  integration  become  in- 
appreciably small  and  can  be  neglected,  both  the  current 
and  the  charge  are  simple  harmonic  functions  and  may 
either  lag  behind  or  advance  ahead  of  the  impressed 
E.  M.  F.  The  current  lags  behind  the  impressed  E.  M.  F., 

when  L  GO  >  -7= — ,  and  advances  ahead  of  it  when  L  GO  < 


S~^  •     bKUV*     CKV*   V  lVJ.J.V>V^k?     C*)JULVyl*VL     V/J.     X  V      VV  A-LV^-JJ.     ^J    Ud/       ^s.          ^-j 

COO  C  GO 

When  L  GO  =  — — ,  that  is,  when  GO  =       ^  ,  there    is   no 

lag  or  advance,  and  the  current  is  exactly  in  phase  with 
the  impressed  E.  M.  F.  In  this  case  the  current  equation 
becomes 

.      E    . 

i  =  -ft  sin  GO  t, 

which  is  identical  with  the  current  equation  obtained  from 
Ohm's  law,  without  considering  either  self-induction  or 


RESISTANCE,    SELF  INDUCTION,   AND   CAPACITY.    131 

capacity.     When  the  sine  is  unity  in  (181),  the  maximum 
value  of  the  current,  represented  by  /,  is 

(187)  1 


From  the  analogy  of  this  equation  to  Ohm's  law,  we  see 

that  the  expression  \  IF  -f-  ( -^ L  GO  )   is  of  the  nature 

of  a  resistance,  and  is  the  apparent  resistance  of  a  circuit 
containing  resistance,  self-induction,  and  capacity.  This 
expression  would  quite  properly  be  called  "impedance," 
but  the  term  impedance  has  for  several  years  been  used  as 
&  name  for  the  expression  VR*  +  L*  o>a,  which  is  the  appa- 
rent resistance  of  a  circuit  containing  resistance  and  self- 
induction  only  [see  equation  (29),  Chapter  III.].  We  would 
suggest,  therefore,  that  the  word  "impediment"  be  adopted 


as  a  name  for  the  expression  V  R?  +  [75 —  —  L  GO)  ,  which 

*  \j  GO  I 

is  the  apparent  resistance  of  a  circuit  containing  resistance, 
self-induction,  and  capacity,  and  that  the  term  impedance 
be  retained  in  the  more  limited*  meaning  it  has  come  to 
have,  that  is,  VR*  -f-  Z2  or*,  the  apparent  resistance  of  a  cir- 
cuit containing  resistance  and  self-induction  only.  Equa- 
tion (187)  may  be  written 

Maximum  E.  M.  F. 

(188)  Maximum  current  =  ^ j-. — 

Impediment 

Since  the  virtual  current  (the  square  root  of  the  mean 
square  of  the  instantaneous  values  of  the  current)  is  equal 

to  -—  times  the  maximum  value  of  the  current,  and  since 

vz 

the  virtual  E.  M.  F.  —  --=  times  the  maximum  E.  M.  F., 

4/2 

Virtual  E.  M.  F. 

(189)  virtual  current  =  — ^ T- — 

Impediment 


132  CIRCUITS  CONTAINING 

It  is  convenient  to  consider  the  impediment  as  a  resistance, 
and  we  are  justified  in  so  doing  inasmuch  as  it  has  the 
same  dimensions  as  a  resistance,  that  is,  a  velocity  in  the 
electromagnetic  system  of  units. 

2  ir 

00  = 


Time 
L  =  Length. 

Therefore,  LGO  =  -~^  J-  =  velocity. 


Length 

1         Length 

-75 —  =  -7^7 =i  velocity. 

CGO         Time 

This  gives  the  dimensions  of  a  velocity  to  the  whole  expres- 
sion for  the  impediment,  which  may  therefore  be  considered 
as  a  resistance. 

The  several  particular  cases  of  circuits  containing  vari- 
ous combinations  of  resistance,  self-induction,  and  capacity 
may  readily  be  found  by  means  of  the  general  solution, 
equation  (181). 

CASE  A.    CIRCUITS  CONTAINING  EESISTANCE  AND  SELF- 
INDUCTION  ONLY. 

.  In  this  case  the  circuit  has  resistance  R  and  self-induc- 
tion Z,  and  an  harmonic  E.  M.  F.,  E  sin  GO  t,  There  being 
no  condenser  in  the  circuit,  the  capacity  C  is  infinite  [see 
page  67,  Chapter  IV.].  After  the  lapse  of  a  very  small  time 
the  terms  containing  the  constants  of  integration  in  the 
general  solution  may  be  neglected  as  explained  above. 
Substituting  in  (181)  C—  oo  ,  we  have 

i  =  — 2  =  sin   •!  GO  t  —  tan'1  — 75-  [• . 

A/  £>2      I        7~2  /~i2  JLl       \ 

V  -K    "T"  -L-J    GO  \ 


RESISTANCE,   SELF  INDUCTION,   AND  CAPACITY.     133 

This  equation  has  been  independently  obtained  from  the 
differential  equation  [see  equation  (28),  Chap.  III.].  The 
current  must  always  lag  behind  the  impressed  E.  M.  F.  by 

Loo 
an  angle  whose  tangent  is  — -  .     In  this  case  the  impedi- 


ment takes   the    particular   value   V R*  -|-  D  c»a,  which  is 
known  as  the  impedance  of  the  circuit. 

CASE  B.    CIKCUITS   CONTAINING    EESISTANCE   AND   CAPACITY 

ONLY. 

In  this  case  the  circuit  has  resistance  R  and  capacity 
(7,  with  an  harmonic  E.  M.  F.,  e  =  E  sin  GO  t.  Substituting 
Z  =  0  in  the  general  equation  (181),  we  have 


sin    Got  +  tan 

( 


This  equation  has  been  independently  obtained  from  the 
differential  equation  [see  equation  (78),  Chapter  V.].  The 
current  must  always  advance  ahead  of  the  impressed 
E.  M.  F.,  when  there  is  resistance  and  capacity  only  in  the 

circuit,  by  an  angle  whose  tangent  is  ~~B  — 


CASE  C.    CIKCUITS  CONTAINING  EESISTANCE  ONLY. 

In  this  case  the  self-induction  L  —  0,  and  the  capacity 
C  =  oo  .  Substituting  these  values  in  the  general  solution 
(181),  we  have 

.      E 

i  =  -„  sm  GO  t. 


134  CIRCUITS  CONTAINING 

This  result  is  immediately  derivable  from  Ohm's  law.  Thus, 
Since     e  =  E  sin  GO  t, 
e       E   . 


E    . 
or        i  —  -     sin  GO  t. 


CASE  D.    CIRCUITS  CONTAINING  CAPACITY  ONLY. 

In  this  case  R  =  0,  and  L  =  0.  Substituting  in  the 
general  equation  (181),  we  have 

i  =  C EGO  sin  \  cot  +  o 
(  * 

This  is  identical  with  equation  (80),  Chapter  V. 

EFFECTS  OF  VARYING  THE  CONSTANTS  OF  A  CIRCUIT. 

The  general  equation  (181)  enables  us  to  ascertain  the 
current  which  will  flow  in  a  circuit  when  we  know  its  re- 
sistance, self-induction,  and  capacity,  the  value  of  the  im- 
pressed E.  M.  F.  and  its  frequency.  It  is  important  to  know 
two  things  about  the  current ;  first,  its  maximum  value  /, 
and,  second,  the  angle  6  by  which  it  lags  behind  or  advances 
ahead  of  the  impressed  E.  M.  F.  The  mean  square  value 
is  readily  obtained  from  the  maximum  value.  We  are  given 
7?,  C,  L,  E,  and  GO.  The  angle  of  lag  or  advance  is 


(190) 


...          1  Lv 

or         tan  9  = 


This  is  an  angle  of  advance  or  lag,  according  as 


GO 


RESISTANCE,    SELF  INDUCTION,   AND   CAPACITY.     135 

greater  or  less  than  -4^-  .     The    maximum  value    for   the 
current  is  . 

E. 

7TT  7">  Tjl 

(191)  7=  —  ,  =  --  -  -5  cos  8. 

r  ' 

-  Lca 


It  is  interesting  to  note  how  any  change  in  7?,  Z,  (7,  G?,  or 
E  affects  the  value  of  6  and  the  current. 

First.  If  the  impressed  E.  M.  F.  E  is  varied,  and  7?,  Z, 
(7,  and  &?  are  maintained  constant,  6  is  not  affected,  and 
the  angle  of  lag  or  advance  remains  unchanged.  The  value 
of  the  current  is  varied  in  direct  proportion  to  E. 

let  E. 

/Second.  If  the  resistance  It  of  the  circuit  is  varied,  and 
Z,  C,  GO,  and  E  are  maintained  constant,  as  E  is  increased, 
the  angle  of  lag  or  advance  is  diminished. 

tan  8  cc  y>  • 

The  sign  of  tan  6  is  positive  or  negative,  and  the  angle 
therefore  one  of  advance  or  lag,  according  to  the  values  of 
Z,  (7,  and  &?,  and  is  independent  of  any  variations  in  the 
resistance.  The  current  is  in  all  cases  diminished  by  an 
increase  of  resistance,  but  the  amount  of  this  decrease 
depends  not  only  upon  R,  but  upon  the  relation  between 


~  Z  GO. 

Coo 

In  Fig.  29  are  shown  two  particular  cases  of  the  varia- 
tion in  the  current  produced  by  change  in  the  resistance. 
Curve  I.  is  for  a  circuit  in  which 

Self-induction  Z  =  2  henrys  =  2  X  109  C.  G.  S.  units. 
Capacity  C  =  .55  microfarads  =  .55  X  10'15  C.  G.  S.  units. 
Impressed  E.  M.  F.  E  =  200  volts  =  200  X  10s  C.  G.  S.  units. 
2  n  n  =  GO=  955. 


136 


CIRCUITS  CONTAINING 


The  abscissae  represent  resistance  in  ohms  (1  ohm  =  109 
C.  G.  S.  units  of  resistance).  The  ordinates  represent  cur- 
rent in  amperes  (1  ampere  =  10'1  C.  G.  S).  units.  The 


FIG.  29. — VARIATION  OF  CURRENT  WITH  CHANGE  IN  RESISTANCE  IN  A 
CIRCUIT  IN  WHICH  E  =  200,  G  =  .55,  L  =  2. 

relation   between   Z,    (7,  and    co  here   taken   is  such  that 


Tr  =  L  co,  or  GO  = 


.—_. 
L  C 


,  which  is  the  relation  that  gives 


no  angle  of  lag  or  advance.  The  relation  between  current 
and  resistance  is  the  same  as  in  Ohm's  law,  and  when 
plotted  gives  the  hyberbola  curve  I.  In  the  same  figure, 
curve  II.  represents  the  value  of  the  current  with  'different 
resistances  in  a  circuit  in  which 


L  =  2  henrys, 

C  =  .55  microfarads, 


E  —  200  volts, 

GO  =  either  1000  or  912. 


The  constants  here  are  the  same  as  in  the  previous  case, 
with  the   exception   of  G?,  which  has  been  changed  from 

955,  that  is,  — == ,  to  either  1000  or  912.     Any  change  in 

V  L  G 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.     137 

GO  from  the  value  — ,  whether  it  be  an  increase  or  de- 

VL  C 

crease,  causes  the  curve  to  depart  from  the  hyperbola,  curve 
I.  It  is  to  be  noticed  that  a  change  in  frequency  of  only 
seven  alternations  per  second  will  change  the  curve  from  I. 
to  II. 

Third.  If  the  coefficient  of  self-induction  L  is  varied 
while  Ry  C,  GO,  and  /Tare  maintained  constant, 

When  L  <  -^—^ ,  tan  0  is  positive  and  0  is  an  angle  of  ad- 

C  GO 

vance. 

0  becomes  less        )          T  . 

\  as  L  increases. 

1  becomes  greater  ; 

When  L  >  ~ — 5-  ,  tan  0  is  negative  and  0  is  an  angle  of  lag. 

\_/  GO 

0  becomes  greater  )          T  . 

\  as  L  increases, 
/becomes  less         ) 

These  changes  in  the  angle  of  lag  or  advance  and  the  cur- 
rent, due  to  change  in  the  self-induction,  are  better  seen 
from  the  consideration  of  a  particular  case.  In  Fig.  30  the 
values  of  0  and  /  are  plotted  for  various  values  of  L  in  a 
circuit  in  which 

E  —  50  ohms,  GO  =  1000, 

C  =  .55  microfarads,  E  =  200  volts. 

When  L  =  -^ — ;  =  1.82,  the    current  has    its    maximum 

C  GO 

value  equal  to  -^ ,  and  0  =  0.     This  is  a  critical  point,  and 

a  slight  change  of  L  in  either  direction  will  cause  0  to  reach 
a  considerable  value  and  the  current  to  fall  to  a  small  part 
of  the  maximum  value.  If  L  be  increased  from  1.82  to 
1.92,  0  changes  from  zero  to  —  63°,  an  angle  of  lag,  and  the 
current  falls  from  4  to  1.8  amperes.  If  L  be  made  1.72, 


138 


CIRCUITS  CONTAINING 


6  becomes  an  angle  of  advance  of  63°  and  the  current  will 
be  1.8  amperes.  It  is  thus  seen  that  an  exact  balance  of 
self-induction  and  capacity  would  be  exceedingly  hard  to 
maintain  in  this  case,  for  a  slight  change  in  the  self-induc- 
tion would  cause  a  large  angle  of  lag  or  advance  and  a  large 
diminution  in  the  current.  Just  how  critical  the  curves 
will  be  in  the  vicinity  of  the  point  of  equilibrium  depends 


90 


FIG  30. — VALUE  OF  CURRENT,  AND  ANGLE  OF  ADVANCE  OR  LAG  FOR 
DIFFERENT  AMOUNTS  OF  SELF-INDUCTION  IN  A  CIRCUIT  IN  WHICH 
R  =  50,  C—  .55,  E  =  200,  GO  =  1000. 

upon  the  constants  of  the  circuit.  The  curves  will  always 
be  of  a  form  similar  to  those  in  Fig.  30,  but  will  often  be 
decidedly  modified  by  the  particular  values  of  72,  (7,  and  GO. 
The  critical  parts  of  the  curves  may  be  more  or  less  marked 
according  to  these  particular  values. 

Fourth.  If  the  capacity  G  is  varied  while  7?,  Z,  GO,  and 
E  are  maintained  constant. 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.     139 

When  C  <  j—, ,  tan  6  is  positive,  and  0  is  an  angle  of  ad- 
vance. 


0  becomes  less 
/becomes  greater 

1 


j- 


as  C  increases. 


When  G  >  -j—^  ,  tan  6  is  negative  and  0  is  an  angle  of  lag. 

0  becomes  greater  )          ~  . 

r.  f  as  C  increases. 

/  becomes  less 

These  changes  of  current  and  lag,  with  the  variation  in 
capacity,  are  shown  in  Fig.  31  for  a  particular  case  in  which 


R  =  50  ohms, 
L  =  2  henrys, 


oo  =  1000, 
E  =  200  volts. 


FIG.  31. — VALUE  OF  CURRENT,  AND  ANGLE  OP  ADVANCE  OR  LAG  POK 
DIFFERENT  CAPACITIES  IN  A  CIRCUIT  IN  WHICH  R  —  50,  L  =  2, 
E  =  200,  GO  =  1000. 


140  CIRCUITS  CONTAINING 

The  maximum  value  for  the  current  occurs  when  C  =  -7 — .,  = 

JLy  GO" 

.5  microfarads.  This  is  a  critical  point  in  the  curve  similar 
to  that  in  the  curves  where  the  self-induction  was  varied. 
Here  6  is  zero,  and  so  the  current,  being  in  phase  with  the 
impressed  E.  M.  F.,  has  a  value  of  4  amperes  in  accord- 
ance with  Ohm's  law.  The  critical  nature  of  the  curves 
here  is  seen  by  the  fact  that  when  C  =  .55  there  is  an  angle 
of  lag  of  75°  and  1=  1.07  ;  when  C  =  .458,  there  is  an  angle 
of  advance  of  75°.  When  6yis  changed  from  .5  to  .488,  the 
current  falls  from  4  to  2.83  amperes  and  is  put  45°  out  of 
phase  in  advance  of  the  E.  M.  F. 

Fifth.  If  the  frequency  is  varied  while  R,  C,  Z,  and  E 
are  maintained  constant,  still  more  marked  changes  occur 
in  the  values  of  /  and  0. 

When  GO  <      ,  tan  0  is  positive  and  0  is  an  angle  of 

\  LC 


advance. 

6  becomes  less 
1  becomes  greater 


j-  as  GO  increases. 


When  GO  >  —     — ,  tan  6  is  negative  and  6  is  an  angle  of  lag. 

r    ://G 

#  becomes  greater  ) 

f  as  GO  increases, 
/becomes  less 

In  Fig.  32  the  values  of  the  current  and  angle  of  lag  are 
shown  for  different  values  of  GO  in  a  circuit  in  which 

R  =  50  ohms,  C  —  .55  microfarads, 

L  =  2  henrys,  E  =  200  volts. 

When  GO  =  — -=-  =  955,  the  current  has   its    maximum 
VL  C 

value  of  4  amperes,  in  accordance  with  Ohm's  law.  Here 
0  =  0.  A  change  of  five  per  cent,  one  way  or  the  other  in 
this  critical  value  for  GO  causes  an  angle  of  lag  or  advance 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.     141 


of  75°,  and  the  current  falls  to  one-fourth  of  the  maximum. 
Just  how  critical  the  curves  are,  in  the  vicinity  of  this  point 


90 


60" 


30 


00 


ng.le  or  Advance 


">  2 


FIG.  32 — VALUE  OF  CURRENT  AND  ANGLE  OP  ADVANCE  OR  LAG  FOR 
DIFFERENT  FREQUENCIES  IN  A  CIRCUIT  IN  WHICH  21  =50,  L  =  2, 
C  =  .55,  E  =  200. 

of  equilibrium  depends  upon  the  particular  values  of  R,  C, 
and  L. 

In  Fig.  33  is  shown  the  E.  M.  F.  necessary  to  cause  a 
constant  current  to  flow  in  a  circuit  in  which  R,  C,  and  GO 
are  constant.  In  the  particular  case  plotted, 


R  =  50  ohms, 

C  =  .55  microfarads, 


I  =  1  ampere, 
GJ  =  1000. 


As    the    self-induction    is    increased     up     to    the    value 


T    


G&o 


-9  =  1.82,  the  E.  M.  F.  needed  to  drive  the  current 


becomes  less  and  less,  and  when  L  =  1.82  the  E.  M.  F. 
needed  is  only  50  volts.  As  L  increases  past  this  critical 
value,  the  value  of  the  E.  M.  F.  needed  increases.  Except 


142 


CIRCUITS  CONTAINING 


very  near  the  critical  point,  tlie  change  in  the  necessary 
E.  M.  F.  is  almost  directly  proportional  to  the  change  in 
the  self-induction,  that  is,  the  curve  is  formed  of  two  straight 
lines  with  a  rounded  point.  This  curve  is  the  reciprocal  of 


/ 

1500. 

1000. 

1 
600. 

0 

\ 

/ 

\ 

/ 

\ 

f 

\ 

/ 

\ 

\ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

V 

1 

s 

/ 

\ 

[ 

^ 

J 

0                                   1.                                  a.                                 3.         Henrys             4 

FIG.  33. — RELATION  BETWEEN  IMPRESSED  E.  M.  F.  AND  SELF-INDUCTION 
WHEN  1  AMPERE  PLOWS  IN  A  CIRCUIT  IN  WHICH  R  =  50,  C  =  .55, 

a>  =  1000. 

the  corresponding  curve  for  current,  with  E  constant  and 
L  variable,  as  shown  in  Fig.  30. 

THE  ENERGY  EXPENDED  PER  SECOND  UPON  A  CIRCUIT  IN  WHICH 
AN  HARMONIC  CURRENT  is  FLOWING. 

The  energy  expended  in  any  circuit  in  the  time  dt  is 
the  product  of  the  E.  M.  F.  and  current  at  that  instant  by 
the  time ;  that  is, 

dW=eidt.     [See  equation  (5),  Chap.  L] 

When  the  E.  M.  F.  is  harmonic  the  instantaneous  value  of 
it  is  e  =  E  sin  cot.     The   current  at  the  same  instant  is 
i  =  /sin  \oot  —  6\.     Therefore  the  differential  equation  of 
energy  is 
(192)  dW=  EIsm  cot  sin  {cot  —  6\dt. 


RESISTANCE,  SELF  INDUCTION,  AND   CAPACITY.    143 

Integrating  between  the  limits  zero  and  T,  the  time  of  one 
complete  period,  we  obtain 


(193)  W=EI       sin  cot  sin  {cot -0}dt. 

VQ 

Expanding  sin  {  GO  t  —  6\,  we  obtain 

/»r  nT 

W  —  El  cos  6  I     sin*  ootdt  —  EI$m6       sin  GO  t  cos  GO  t. 
«/o  t/o 

Keplacing  sin  GO  t  cos  a?  £  by  its  equivalent  -:        — , 

a 

pT   .                    EI  sin  9  /»r  , 
W=Elcos&  I     sin* cot dt  — ^ /    sin%Gotdt. 

t/O  ^  «^0 

Between  the  limits  zero  and  T  the  second  integral  vanishes 

T 

and  the  first  integral  is  equal  to  -^-.     [See  page  37.]     The 

value  of  the  energy  expended  per  period  is  therefore 


(194)  W 

The  energy  expended  per  second  is  therefore 

(195)  -,:, >;     F 

that  is,  the  energy  per  second  is  half  the  product  of  the 
maximum  E.  M.  F.  by  the  maximum  current  by  the  cosine 
of  the  angle  of  difference  between  the  E.  M.  F.  and  cur- 
rent. Since  the  effective  E.  M.  F.  or  current  is  equal  to 

—=•  times  the  maximum  value,  we  have 

(196)  W=Efeose, 

meaning  by  E  and  /the  square  root  of  the  mean  square 
values  of  E.  M.  F.  and  current. 


CHAPTER  X. 

CIRCUITS  CONTAINING  RESISTANCE,  SELF  INDUCTION,  AND 

CAPACITY. 

CASE  III.    (CONTINUED.)     CURRENTS  AT  THE  "MAKE"  FOR 
AN  HARMONIC  E.  M.  F. 

CONTENTS: — Complete  equations  for  i  and  q  with  the  complementary 
function  in  the  oscillatory  form.  To  determine  the  constants  A'  and 
$'.  To  determine  the  constants  A  and  <?.  Complete  equation  for  ^ 
with  constants  determined.  Examples  to  explain  the  general  equation 
in  cases  of  particular  circuits.  Curves  showing  the  current  at  the  make 
for  a  particular  circuit.  The  phase  at  which  the  E.  M.  F.  should  be 
introduced  to  make  the  oscillation  a  maximum. 

IN  the  discussion  of  the  current  equation  in  Chapter 
IX.  for  an  harmonic  E.  M.  F.,  it  was  stated  that  after  the 
lapse  of  a  very  short  time  the  exponential  terms,  equation 
(181),  become  inappreciably  small  and  can  be  neglected, 
and  the  discussion  of  the  equation  there  given  only  applies 
after  the  current  has  been  flowing  for  a  short  time.  It  is 
proposed  in  this  chapter  to  investigate  the  effect  of  these 
exponential  terms  in  modifying  the  current  during  the  very 
short  time  after  the  "  make,"  or,  in  other  words,  after  the 
harmonic  E.  M.  F.  is  suddenly  introduced  into  the  circuit. 
The  E.  M.  F.  may  be  introduced  at  any  point  of  its  phase, 
that  is,  it  may  be  zero  or  may  have  its  maximum  or  any 
intermediate  value,  but,  in  any  case,  the  complete  equations 
(181)  and  (182)  show  just  what  happens,  provided  we  de- 
termine the  constants  cl  and  ca  of  the  complementary  func- 

144 


UESISTANCE,  SELF  INDUCTION,  AND   CAPACITY.     145 

tion,  so  that  they  correspond  to  the  particular  hypothesis 
made. 

It  has  been  noted  (120)  that  the  complementary  func- 

.*  _  — 

tion  c,  e    Tl  +  ca  e    T*  may  be  written  in  another  form,  viz. : 


Rt 


-  I?  C* 


2LC 


This  latter  form  must  be  used  when  we  have  the  relation 
4Z  >  R*C,  for,  under  this  hypothesis,  the  time-constants 
Tl  and  T^  of  the  first  form  become  imaginary.  To  make 
this  supposition  is  equivalent  to  saying  that  the  character 
of  the  discharge  from  the  circuit  is  oscillatory  [see  Chap- 
ter VII.].  Inasmuch  as  this  relation  4Z  >  IFCis  true  for 
most  ordinary  circuits  in  which  L  has  an  appreciable  value, 
and  since  the  results  obtained  are  rather  more  interesting 
under  this  supposition  than  under  the  supposition  that 
4Z  <  7?2<7,  which  would  give  "dead  beat"  discharge,  we 
will  confine  our  attention  to  the  oscillatory  case  only.  The 
plan  to  be  followed  in  the  discussion  of  this  subject  will  be 
to  determine  the  constants  A  and  $  of  the  general  equa- 
tion, and  write  the  general  result.  The  application  of  this 
result  to  a  particular  circuit  will  then  be  made,  and  curves 
drawn  showing  the  current  as  it  starts  in  this  circuit  before 
it  has  reached  its  final  harmonic  form. 

The  general  equation  for  current,  under  the  assumption 
made  that  4Z  >  ./22(7,  may  be  written 

(197)   i  =  -7=    =  f  sin     <.  t  +  tan 


146  CIRCUITS  CONTAINING 

where  A  and  $  are  the  constants  of  integration  to  be  deter- 
mined and  are  each  of  them  real.  Likewise,  the  equation 
expressing  the  quantity  of  charge  on  the  condenser  at  any 
moment  may  be  written  [see  (182)  and  (123)] 

—  E 

(198)     q  =  -  ---  =    =*  cos  j  c 


tan 


-1 


B* 


To  determine  the  constants  A'  and  $' ; — Kemembering  the 
relation  d  q  =  i  d  t,  we  may  differentiate  (198)  and  write 


m 

2L 


.  A  ,    ,.,  . 

Sln-  -""'*     tan 


Equating  (199)  with  (197),  we  obtain  the  relations 
(200)  :.        •    -A^^=,  •        '/ 


(201) 


For  simplification  make  the  following  substitutions : 
(202)        1=  ^         ==,.        [See  (191).] 


RESISTANCE,  SELF  INDUCTION,  AND   CAPACITY.     147 

(203)        *  =  cat  +  tan-1  ~  -  ~        =  °»t  +  0. 


(204) 


The  frequency  of  oscillation  is  ~  —  >  and  the  period  —  . 

Then  we  may  write,  after  substituting  in  (197)  and  (198) 
the  values  of  A'  and  $>'  as  determined, 


R_t 
2~L 


(205)    •*'  =  fsinifi  +  Ae   '  ^  sin  {at  +  $}. 


r  _*< 

(206)      q=  —  -  -  --' -   '     "'^"- 


sin  )  at  4-  ^  -I-  tan~: 


7b  determine  the  constants  A  and  $: — In  these  equations 
time  is  counted  from  the  point  when  the  impressed  E.  M.  F. 
is  zero.  Let  tl  be  the  time  when  the  E.  M.  F.  is  introduced. 
We  know  then  that  the  current  and  the  charge  of  the  con- 
denser are  each  zero  at  the  time  tl ,  the  condenser  having 
no  initial  charge.  These  conditions  alone,  namely,  that 
i  =  0  and  q  =  0  when  t  =  tt ,  are  sufficient  to  determine  the 
constants.  In  equations  (205)  and  (206)  make  i  =  0,  q  —  0 
when  t  =  tl ,  and  call  fyl  the  value  of  if>  when  t  =  tl ,  and  we 
have 

_^ 
(207)    Oss 


(208)    0  =  -  ^  cos  fr  +  A  VL  Ce   ~2L 


148  CIRCUITS  CONTAINING 

Eliminating  A  between  these  equations,  we  obtain 


Substituting  this  value  of  $  in  (207), 
(210)       A  =  -Ie+ 


smcot  - 


This  expression  for  A  may  be  reduced  by  simple  trigo- 
nometrical operations  to  the  form 


,Rt_ 

(211)    A~-         ^ 


V±LG-> 

V(L  Coo*  -  1)  sin2  ^  +  %  E  C GO  sin  2^  +  1. 

Substituting  these  values  of  A  and  $  in  equation  (197),  we 
may  write  the  complete  solution  with  constants  determined, 


. 


y  4  X  (7  — 


f  2cot^1  +  (7^0^-1  ) 

sin  J  *(<-—t)+  cot"1  —     -  f   ri    '       -^^      V  » 

d7-^a6Y«J  ) 


There  are  several  general  conclusions  which  can  be  made 
in  interpreting  the  meaning  of  this  equation.  It  is  evident 
that  there  will  be  an  oscillation  of  the  current  when  the 
E.  M.  F.  is  first  introduced,  which  gradually  dies  away,  the 
rate  of  dying  away  depending  upon  the  exponent  of  e  in  the 
equation  or,  in  other  words,  upon  the  time-constant  of  the 

2  L 
circuit,  namely,  -75-  *     The  initial  value  of  this  logarithmic 

JK 

decrement   curve,  that  is  at  the  make  when  t  =  tl  ,  is  ex- 
pressed by  the  coefficient  of  e  in  the  equation.   It  is  evident 


RESISTANCE,  SELF  INDUCTION,  AND  CAPACITY.    149 

that  this  initial  value  depends  upon  the  value  of  $l  for  its 
value,  or  is  a  function  of  ^\.  The  initial  value  of  the  log- 
arithmic curve  has,  then,  a  different  value  for  every  value 
of  i/>lt  i.e.,  for  every  point  of  the  phase  of  what  the  current 
would  have  been  if  it  had  started  out  at  the  make  as  an 
harmonic  current  having  the  same  phase  difference  with 
the  E.  M.  F.  as  it  finally  assumes.  Again,  at  the  time  t  —  tl 
the  value  of  the  last  term  of  the  equation  becomes  —/sin  0,. 
This  will  be  evident  upon  replacing  the  coefficient  of  e  by 
its  value  given  in  (210).  The  first  term  becomes  /sin  ^, , 
when  t  —  t1 ,  and  the  two  terms  together  show  that  the 
equation  makes  the  value  of  the  current  zero  at  the  time  tlt 
that  is,  at  the  time  the  E.  M.  F.  is  introduced. 

In  order  to  show  the  meaning  of  this  equation  more 
clearly,  a  particular  example  will  be  assumed.  Suppose 
we  have  a  circuit  with  a  resistance  of  50  ohms,  a  self-induc- 
tion of  2  henrys,  and  a  capacity  of  .55  microfarads,  all  in 
series.  Such  a  circuit  would  correspond  nearly  to  the  fine 
wire  coil  of  a  small  10-light  Westinghouse  transformer 
connected  in  series  with  a  condenser  of  .55  microfarads 
capacity.  Let  an  E.  M.  F.  of  100  volts  (maximum  value), 
having  a  periodicity  of  159,  be  impressed  upon  the  circuit ; 
that  is,  the  angular  velocity  GO  =  2  n  X  159  =  1000,  approx- 
imately. We  have,  then,  with  these  values  assumed, 

E  =  100  volts  (max.)  =  100  X  108     C.  G.  S.  units. 
R  =  50  ohms  =    50  X  109    C.  G.  S.  units. 

/  =  2  henrys  =       2  X  109     C.  G.  S.  units. 

C  =  .55  microfarads   =  .55  X  10-15C.  G.  S.  units. 
GO  =  1000. 

f^$M^    =  -08  seconds. 
/  —  .53  amperes  (max,).     [See  equation  (202).] 
6    =  -  74°  30'. 

a   =  955  =  2  n  X  frequency  of  oscillation  =  2  n  x  151. 
[See  (204).] 


150  CIRCUITS  CONTAINING 

The  equation  for  the  current  in  this  particular  case  is 


(213)     i  =  .53  sin  0  -  .477  V.I  sin*  fa  +  .0137  sin  2  fa  +  1 


.08 


sin  J955(£  — 


Curve  III.,  Fig.  34,  represents  the  plot  of  this  equation 
when  the  particular  value  of  ij\  ig  30°.  This  means  that 
the  E.  M.  F.  is  introduced  into  the  circuit  at  that  particular 
time  at  which  the  normal  current  curve  is  30°  from  its  zero 
value.  The  value  of  the  coefficient  of  e  when  ip1  is  30°  is 
.495,  and  the  equation  reads 


(214)      i  =  .53  sin  $  —  .495  e     -08   sin  j  955  (t  —  tj  +  x}- 

Here  x  stands  for  angle  of  lag  expressed  in  equation  (212), 
and  is  not  expressed  in  figures  inasmuch  as  it  is  not  neces- 
sary to  know  it  in  order  to  draw  the  curves,  because  the 
phase  is  determined  by  the  fact  that  we  know  the  distance 
0'  Af,  Fig.  34,  it  being  equal  but  of  opposite  sign  to  the 
distance  0  A.  It  will  be  noticed  that  the  initial  value  of 
the  logarithmic  decrement  is  nearly  the  same  for  any  value 
of  ^\  in  this  particular  case.  Moreover,  as  it  happens,  the 
initial  value  of  the  logarithmic  decrement  is  nearly  the 
same  as  the  maximum  value  of  the  current  I.  Curve  I.  is  a 
sine  curve  representing  the  first  term  in  equation  (214),  and 
curve  II.  a  sine  curve  with  logarithmic  decrement  repre- 
senting the  second  term  in  the  equation.  The  current 
curve,  III.,  is  the  sum  of  curves  I.  and  II.  After  about 
one-tenth  of  a  second,  curve  II.  becomes  inappreciable  and 
the  current  follows  a  simple  sine  curve. 

As  a  second  example,  let  us  consider  the  same  circuit 
as  before.  But  now  suppose  the  frequency  is  just  half 
what  it  was  in  the  first  example,  namely,  79.5,  or  that 
CL>  =  500.  Furthermore,  suppose  the  E.  M.  F.  is  such  that 
it  will  send  a  maximum  current  of  .5  of  an  ampere  through 


RESISTANCE,  SELF  INDUCTION,  AND   CAPACITY.    151 


152  CIRCUITS  CONTAINING 

the  circuit.  It  will  be  found,  upon  calculation,  that  the 
E.  M.  F.  must  be  1320  volts  maximum.  With  these  values, 
then, 

E  =  1320, 

R  =  50, 

L  -2, 

C  =  .55, 

GO  =  500, 

T  =  .08  seconds, 

/   =  .5  amperes, 

B    =  88°  55',  tan  6  =  52.8, 

a   =  955, 

the  equation  for  the  current  becomes 


(215)  i  =  .5  sin  ^  —  .955  V-  .725  sin2  ^  +  .0069  sin  2^ 


.08 


sin  J955(£- 


The  plot  of  this  equation,  when  ^  is  taken  equal  to  180° 
(that  is,  the  E.  M.  F.  is  introduced  when  the  normal  current 
curve  is  zero),  is  shown  in  Fig.  35.  It  will  be  noticed  that 
the  initial  value  of  the  logarithmic  curve  has  considerable 
variation  according  to  the  particular  point  of  time  at  which 
the  E.  M.  F.  is  introduced.  This  variation  is  represented 
in  the  curve  IV.,  Fig.  35.  The  initial  value  of  the  logarith- 
mic decrement  at  0°  or  180°  is  almost  twice  as  much  as  the 

955 
maximum  value  of  the  current  7,  their  ratio  being  :—  p—  . 

The  equation,  when  ipl  is  180°,  reduces  to 


(216)     i  =  .  5  sin  #-.955  e         !    sin  1  955  (^  - 

In  each  of  the  above  examples  the  current  follows  the 
sine  law  in  about  one-quarter  of  a  second  after  the  periodic 
E.  M.  F.  is  introduced,  during  which  time  somewhere  in  the 
neighborhood  of  forty  oscillations  have  been  made. 


RESISTANCE,  SELF  INDUCTION,  AND   CAPACITY.     153 

The  phase  at  ivhich  the  E.  M.  F.  should  be  introduced  to 
make  the  oscillation  a  maximum : — It  may  be  interesting  to 
inquire  at  what  point  the  E.  M.  F.  should  be  introduced 


•< 


g  o 

s  ^ 

§  o 

00  g 


a  S 


e  g 

^       ^j 


o 

-«1     H 

.     O 
M     5 


O     o 

S    h    o 


into  the  circuit  to  render  the  effect  of  the  oscillation  a 
maximum.  This  point  may  readily  be  found  by  referring 
to  equation  (212).  The  coefficient  of  e  becomes  a  maximum 


154 


CIRCUITS  CONTAINING 


(for  a  variation  in  ^),  when  the  quantity  under  the  radical 
sign  is  a  maximum.     Differentiating  the  quantity  under  the 


82  30 


FIG.    36. — SHOWING   HOW   TO   FIND   GEOMETRICALLY   THE   ANGLE   ^ 

WHICH  MAKES  THE  EFFECT  OF  THE  EXPONENTIAL  TEEM  A  MAXIMUM. 


radical,  then,  with  respect  to  tlt  and  equating  to  zero,  we 
obtain 

(217)        (L  Co?  -  1)  sin  2  fa  +  R  CGO  cos  2^  =  0. 


Whence     tan  2  fyl  =  =— : 

But  it  will  be  remembered  that  [see  equation  (190)] 

1  -  L  CGO* 


. 

RESISTANCE,  SELF  INDUCTION,  AND   CAPACITY.    155 


Hence    tan  2^,  =  cot  6  =  tan        —  #> 


(218)  _  , 
And  since  ^  =  G?^  +  #  [see  (138)],  we  find 

(219)  •//.;       <^.=f-T-     -:.'•'       ..;   '       •'..'''. 

Suppose  6  is  an  angle  of  lag  of  —  75°,  as  in  the  first  example 

n        75° 
cited,  then  its  sign  is  negative  and  i/>1  =  j-  +  -~-  =  82°  30' 

for  a  maximum.  If  #  is  -{-  88°  557,  as  in  the  second  exam- 
ple, ^  =  45°  -  44°  27;.5  =  327.5  for  a  maximum. 

The  curve  IV.,  Fig.  35,  shows  that  the  maximum  point 
is  nearly  at  the  position  where  ^  =  0,  and  thus  agrees  with 
this  result.  The  exact  form  which  the  current  curve 
assumes  at  the  introduction  of  an  harmonic  E.  M.  F.  depends 
upon  the  time  of  its  introduction  and  the  constants  of  the 
circuit.  The  curves  shown  in  Figs.  34  and  35  give  an  idea 
of  what  may  be  expected  in  other  cases.  In  all  cases,  after 
a  very  few  periods,  the  current  reaches  the  simple  sine 
form. 

The  current  which  flows  upon  making  a  circuit  which 
contains  resistance  and  self-induction,  but  no  capacity,  is 
shown  in  Fig.  15,  Chapter  III.,  to  which  the  reader  is 
referred. 


CHAPTER  XI. 

CIRCUITS  CONTAINING  RESISTANCE,   SELF-INDUCTION, 
AND  CAPACITY. 

CASE  IV.    ANY  PERIODIC  E.  M.  F. 

CONTENTS:— Fourier's  theorem.  General  equations  for  i  and  q  with  any 
periodic  E.  M.  F.  If  the  self.-induction  and  capacity  neutralize  each 
other  at  every  point  of  time  and  the  current  is  therefore  the  same  as  if 
both  self-induction  and  capacity  were  absent,  the  impressed  E.  M.  F. 
must  be  a  simple  harmonic  E.  M.  F.  If  the  heating  effect,  or  any 
effect  which  depends  upon  ftfdt,  in  a  circuit,  is  the  same  when  the  self- 
induction  and  capacity  are  present  as  it  is  when  they  are  absent,  the 
impressed  E.  M.  F.  must  be  a  simple  harmonic  E.  M.  F.  Various 
types  of  current  curves.  When  curves  are  not  symmetrical,  although 
the  quantity  flowing  in  the  positive  direction  is  equal  to  the  quantity 
in  the  negative  direction,  yet  th&/"&&  effect  will  generally  be  different 
in  these  two  directions.  Illustration  from  a  particular  curve.  Alter- 
nating-current arc-light  carbons. 

IF  we  suppose  that  the  impressed  E.  M.  F.  is  made  up 
of  a  number  of  simple  harmonic  E.  M.  F.'s  added  together, 
the  impressed  E.  M.  F.  may  be  written 

(220)     e  =  E,  sin  (b,  GO  t  +  0,)  +  E,  sin  (&„  a  t  +  0.) 

+  E3  sin  (b3a>t  +  0,)  +  etc. 
and,  therefore, 

de 

-77  =  El  6,  co  cos  (bla)t  +  B,)  +  E^oo  cos  (6a  cot  +  0a)  +  etc. 

QJ  t 

156 


RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY.      157 
Expressed  as  a  summation,  we  have 
(221)  e  =  ^>  E  sin  (b  a?  t  +  0)  = 


fl  r>  ^^mmm 

(222)  IT  =  ®  ^>^&  cos  (ba>t+6)=f'  (t). 

E,b,e 

In  this  summation  it  is  to  be  understood  that  E  and  8  take 
in  succession  any  values,  fractional  or  integral,  but  that  b 
may  only  have  positive  integral  values  as  the  E.  M.  F.  is 
supposed  to  be  periodic,  and  consequently  the  periods  of 
the  component  sine-curves  must  be  commensurable.  It 
was  shown  by  Fourier,  in  his  treatise  on  the  Analytical 
Theory  of  Heat,  published  in  1822,  that  such  an  expression 
as  (220)  or  (221)  represents  any  single-valued  periodic  func- 
tion whatever,  and  is  therefore  an  expression  which  repre- 
sents any  possible  E.  M.  F.  whatever.  If  (222)  is  substituted 
in  the  general  equation  for  current  (99),  and  (221)  in  the 
general  equation  for  charge  (100),  it  will  be  found,  upon 
integrating,  that  each  component  term  in  the  E.  M.  F.  gives 
a  term  in  the  current  or  charge  similar  to  that  given  in 
equations  (181)  and  (182)  in  Case  III.,  and  consequently  the 
resultant  current  may  be  expressed  as  a  summation  thus  : 


(223)    i  =  ^>  — —     ==  ==  sin  J  b  GO  t  +  6 

E,b,e  A  / 7?*  i  ( L_    .  r*«,^          ' 


and  the  charge 


(224)  s<=2-  eogiftgf 

*M6a.   />  +  (     *      .Zj 
y  \Cbos 


1 

+tan 


158  CIllCUITS  CONTAJNINO 

In  these  sums  for  i  and  q  there  must  be  as  many  terms  in 
each  as  there  are  in  the  expression  for  the  E.  M.  F.,  and 
the  values  of  E,  b,  and  0  must  be  the  same  in  corresponding 
terms.  These  equations  express  the  current  and  charge  in 
a  circuit  whose  E.  M.  F.  is  any  periodic  E.  M.  F.,  as  in  equa- 
tion (221). 

If  the  self-induction  and  capacity  neutralize  each  other  at 
every  point  of  time,  and  the  current  is  therefore  the  same  as  if 
both  self-induction  and  capacity  mere  absent,  the  impressed 
E.  M.  F.  must  be  a  simple  harmonic  E.  M.  F.  —  In  the  discus- 
sion of  Case  III.,  where  the  E.  M.  F.  was  harmonic  and  the 
resulting  current  was  shown  to  be  harmonic  also,  it  was 

pointed  out  that  if  the  relation  GO  =  —  -  -    existed,   the 

current  was  the  same  as  if  there  was  no  self-induction  and 
no  condenser  in  the  circuit,  and  the  same  as  if  it  simply 
followed  Ohm's  law.  This  was  shown  by  substituting  the 

relation  GO  —     ,      -  ,  or  77—  —  L  GO  =  0,  in  the  current  or 
V  L  C         ^  <*> 

charge  equations  (181)  and  (182)  and  neglecting  the  com- 
plementary function.  Those  equations,  with  these  substi- 
tutions, become 

E 

i  =  -    sin  GO  t. 


E 
q  =  —  -35  —  cos  co  t. 

-/£  GO 

It  is  seen  that  the  current  and  charge  are  the  same  at 
every  point  of  time  as  if  the  self-induction  and  capacity 
were  absent.  Now,  since  the  current  is  the  same  at  every 
point  of  time,  the  effects  of  this  current  will  be  the  same  ; 
namely,  the  quantity  which  flows  in  a  half  period,  being 

jr 
»a 

/     i  d  t  =  O,  is  tlje  same  as  when  there  is  no  self-induction 

l/a 


RESISTANCE,  SELF-INDUCTION,   AND   CAPACITY.      159 

and  capacity,  and  the  energy  expended  in  the  circuit  in 
performing  work,  or  in  heating  effects,  is  likewise  the  same, 

being  proportional  to  /  i*dt. 

In  order  to  ascertain  whether  some  similar  relation  be- 
tween self-induction  and  capacity  would  cause  them  to 
neutralize  each  other  when  the  impressed  E.  M.  F.  is 
not  a  simple  harmonic  function  of  the  time,  consider  the 
case  where  the  E.  M.  F.  is  composed  of  two  parts,  each  a 
sine-function  of  the  time.  Suppose 

(225)  e  =  El  sin  a  GJ  t  -j-  E^  sin  b  GO  t, 

where  a  and  b  are  integers.  In  the  circuit  there  is  re- 
sistance, self-induction,  and  capacity.  Then  at  any  time 
the  value  of  the  current  is  [see  (223)] 


(226)    i  = 


La  GO—  - 


Catal 


Suppose  the  self-induction  and  capacity  have  the  relation 

a  GO  =.     f          -     Then  they  will  neutralize  each   other  in 
V  J.J  \j 

the  first  term  of  the  above  expression  for  the  instantaneous 
value  of  the  current.     But  in  the  second  term  the  relation 

b  GD  =      is  necessary  to  cause  the  self-induction  and 

V  L  C 


160  CIRCUITS  CONTAINING 

capacity  to  neutralize  each  other.  Now,  if  one  of  the  above 
terms  is  changed  by  the  introduction  of  self-induction  and 
capacity,  while  the  other  term  is  unaffected,  the  value  of 
the  current  which  is  equal  to  the  sum  of  the  two  terms  must 
be  changed.  It  therefore  follows  that  neither  the  relation 

a  GO  —      .         nor  b  GO  = will  cause  the  self-induc- 

VLG  VLC 

tion  and  capacity  to  neutralize  each  other  when  introduced 
into  a  circuit  containing  an  impressed  E.  M.  F.  composed  of 
two  simple  harmonic  E.  M.  F.'s  with  angular  velocities  a  GO 
and  b  GO,  respectively.  If  a  =  b,  the  two  terms  in  the  expres- 
sion for  the  instantaneous  value  of  the  current  may  be  writ- 
ten as  one,  and  we  have  a  simple  harmonic  function  of  the 

time.     The  relation  a  GO  =  boo—  will  then  cause  the 

-self-induction  and  capacity  to  neutralize  each  other. 

If  El  =  0,  or  if  E^  —  0,  then  we  have  a  simple  sine-func- 
tion, and  the  relation  I  GO  =  ,  or  a  GO  =  ,  re- 

v  L  C  v  L  G 

spectively,  will  cause  the  balancing  of  the  self-induction 
and  capacity. 

In  order  to  ascertain  the  conditions  under  which  there 
may  be  self-induction  and  capacity  in  a  circuit,  just  neutral- 
izing each  other,  so  that  the  instantaneous  values  of  the 
current  will  be  the  same  as  though  there  were  no  self-induc- 
tion and  capacity  in  the  circuit,  we  will  consider  the  general 
differential  equation  of  E.  M.  F.'s 


[See  equation  (87).]  We  wish  to  ascertain  the  conditions 
by  which  the  current  will  be  the  same  as  when  there  is 
neither  self-induction  nor  capacity,  that  is,  the  conditions 


" 

RESISTANCE,  SELF-INDUCTION,   AND  CAPACITY.      161 

£> 

by  which  i  =  -^  and  e  =  12i,  according  to  Ohm's  law. 
Substituting  in  the  above  equation,  we  have 

di      fidt 
(227)  £^-  +  --77— =  0. 

This  is  the  same  as  saying  that  the  E.  M.  F.'s  of  self-induc- 
tion and  capacity  are  equal  and  opposite.  By  differentia- 
tion, 

d*  i  idt 

~dt=~  TV' 

Multiplying  by  -77  , 

idi 


LC 

By  integrating  we  have 

(dtJ  ''       ~  ~LC+C* 
di 


The  variables  may  be  readily  separated,  thus : 

(228) 


LC 


The  integral  of  (228)  is  obtained  by  the  formula  of  integra- 
tion, 

C      dx  .x 

I  —  =  sin"1  — . 


162  CIRCUITS  CONTAINING 

Upon  integration  it  becomes 


•   1     *  * 

sin"1  — ==r  =  — ==,  4-  c,. 


Taking  the  sine  of  each  member  and  writing  c'  for  VcL  C,, 
(229)  i  =  c's 

The  only  two  variables  in  this  equation  are  i  and  t,  and 
the  current  is  seen  to  be  a  sine-function  of  the  time.  When 
the  current  is  a  maximum,  the  sine  is  unity  and  we  have 

I=cf. 

If  the  time  is  reckoned  from  the  point  where  the  current  is 
zero,  t  =  0  when  i  =  0,  and  we  have 


Substituting  these  values  for  the  constants  c'  and  c,,  we 
have 

(230)  ; 


In  an  harmonic  function,  as  this,  the  coefficient  of  the  vari- 
able t  is  the  angular  velocity  which  we  designate  by  GO. 
Equation  (230)  then  becomes 

(231)  i  =  /sin  Got. 

We  have,  then,  the  necessary  conditions  by  which  the  self- 
induction  and  capacity  will  just  neutralize  each  other  at 
every  point  of  time.  The  current  must  be  a  simple  sine- 
function  of  the  time,  and  the  self-induction  and  capacity 

must  have  such  values  that  GO  =  -  .    By  no  other  con- 

VL  C 

ditions,  with  self-induction  and  capacity  in  a  circuit,  can 


RESISTANCE,  SELF-INDUCTION,   AND   CAPACITY.      163 

the  instantaneous  values  of  the  current  be  the  same  as 
though  the  capacity  and  self-induction  were  absent. 

If  the  heating  effect,  or  any  effect  which    depends    upon 

/  i*dt,  in  a  circuit  is  the  same  when  the  self-induction  and 

capacity  are  present  as  it  is  ivhen  they  are  absent,  the  impressed 
E.  M.  F.  must  be  a  simple  harmonic  E.  M.  F. — Since  we  have 
found  that  there  is  no  possible  relation  between  L  and  C9 
so  that  the  instantaneous  values  of  the  current  are  unchanged 
by  their  introduction  into  a  circuit  with  an  impressed 
E.  M.  F.  which  is  not  an  harmonic  function,  it  is  interesting 
to  inquire  whether  any  relation  can  be  given  L  and  C  so 
that  the  energy  spent  in  the  conductor  in  a  given  time  is  the 
same  before  as  after  the  introduction  of  L  and  C. 

Before  attempting  to  investigate  such  a  relation,  it  will 
be  well  to  first  consider  some  different  classes  of  current 

curves,  then  ascertain  the   /  i*  d  t  effect  for  some  particular 

current  curves,  and  afterwards  consider  the  energy  of  any 
periodic  curve  whatever. 

Fig.  37  represents  a  curve  which  has  an  equal  area  above 
and  below  the  axis  every  period.     This  means  that  the  in- 


FIG.  37. 


tegral  I  idt  for  one  period  is  zero,  that  is,  the  quantity  of 

-electricity  which  flows  each  period  in  the  positive  direction 
is  equal  to  that  which  flows  in  the  negative  direction. 
Moreover,  if  the  lower  half  of  the  current  curve  is  inverted 


164  CIRCUITS  CONTAINING 

and  represented  by  the  dotted  line,  it  is  an  exact  repetition 
of  the  first  half  of  the  curve.  This  curve  may  represent 
the  type  of  current  curves  given  by  alternating  generators 
in  circuits  with  resistance,  self-induction,  and  capacity ; 
for,  it  is  evident  that,  as  the  armature  revolves,  the  number 
of  lines  introduced  into  the  circuit  every  period  equals 
those  taken  from  the  circuit.  Now,  the  quantity  of  current 
which  flows  is  strictly  proportional  to  the  change  in  the 
number  of  lines  threading  the  circuit.  This  is  equivalent 
to  saying  that  the  quantity  which  flows  in  the  positive  direc- 
tion is  exactly  equal  to  the  quantity  flowing  in  the  negative 
direction,  or  the  total  algebraic  quantity  per  period  is  zero. 
Now,  if  the  generator  is  exactly  symmetrical,  the  current 
curve  in  the  second  half  of  the  period  is,  if  inverted,  an 
exact  repetition  of  the  curve  in  the  first  half.  Any  irregu- 
larities in  the  symmetry  of  the  machine  might  cause  slight 
differences  in  the  two  parts  of  the  curve,  but  hardly  enough 
to  prevent  this  curve  from  representing  the  type  of  curves 
given  by  alternating  machines.  During  every  complete 
revolution  of  the  armature,  the  total  algebraic  quantity  of 
current  flowing  must  be  rigorously  equal  to  zero,  no  matter 
how  many  irregularities  there  may  be  in  the  machine ;  for, 
the  number  of  lines  introduced  into  the  circuit  exactly 
equals  those  subtracted  from  the  circuit,  because  after  a 
complete  revolution  the  number  of  lines  is  the  same  as  at 
the  start.  It  is  possible  that  adjacent  positive  and  negative 
areas  may  be  unequal  in  a  multipolar  machine,  due  to  some 
irregularity  in  the  machine,  but  after  a  complete  revolution 
of  the  armature  the  sum  of  the  positive  areas  equals  the 
sum  of  the  negative. 

Fig.  38  represents  a  current  curve  which  has  equal  areas 
above  and  below  the  axis  every  period,  but  the  negative 
area,  when  inverted,  is  not  necessarily  a  repetition  of  the 
positive  area.  This  represents  the  type  of  current  curve 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.     165 

when  there  is  a  non-leaky  condenser  in  the  circuit,  since 
the  total  algebraic  flow  here  is  necessarily  equal  to  zero. 


FIG.  38. 

Fig.  39  represents  a  current  curve  in  which  the  negative 
area  is  neither  equal  to  the  positive  area  nor  symmetrical 
with  it  when  inverted. 


FIG.  39. 

It  is  interesting  to  inquire  whether  the   /  i*  d  t  effect  is 

the  same  in  a  circuit  while  the  current  flows  in  the  positive 
direction  as  it  is  while  flowing  in  the  negative  direction. 
We  can  see  that  it  is  the  same  for  a  current  of  the 
type  represented  in  Fig.  37,  for,  squaring  the  ordinate  at 

each  point  and  drawing  a  new  curve,  b,  Fig.  40,  the  /  i*  d  t 

effect  is  proportional  to  the  areas  of  this  new  curve.  Since 
the  current  curves  a,  a  are  exact  repetitions,  these  areas,  b,  b, 

are  identical,  and  the   /  i*  d  t  effect  is  the  same  when  the 

current  is  positive  as  it  is  when  negative. 

Let  us  inquire  how  this  is  for  a  current  of  the  type  of 


166 


CIRCUITS  CONTAINING 


Fig.  38,  where  the  areas  are  equal,  that  is,  the  jidt  is  the 
same  for  positive  as  for  negative  current,  but  the  negative 


FIG.  40. 


part,  when  inverted,  is  not  an  exact  repetition  of  the  positive 
part.     In  Fig.  41  the  areas  between  the  axis  and  the  cur- 


^ v 


/     ^-ar^L  \ 


/    \ 


FIG.  41. 


rent  curve  a,  a  are  equal  for  each  half  period.     The  curve 
b,  b  is  drawn  by  squaring  each  ordinate  of  the  curve  a.   The 

areas  b,  b  represent  the   A2  d  t  effect,  and  we  wish  to  find 
whether  they  are  equal. 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.     167 

Illustration  from  a  Particular  Case. — To  show  that  this 

I  i*dt  effect  is  not  necessarily  the  same  when  the  current 

is  positive  as  when  it  is  negative,  it  will  suffice  to  take  one 


-Y3- 


Time 


FIG.  42. 

particular  case  of  a  current  curve.     Suppose  the  positive 
curve  is  a  parabola  (Fig.  42)  whose  equation,  referred  to 

0  as  an  origin,  is 

(232)  *"  =  — 3t  +  3. 

Stippose  that  the  negative  curve  is  a  sine-curve  whose  equa- 
tion, referred  to  0"  as  origin,  is 

(233)  i  =  |  V3  sin  t. 

It  is  easily  shown  that  the  areas  of  these  curves  are  equal. 
Area  parabola  =  f  [base  X  height]. 

One-half  of  the  base  of  the  parabola  is  found  by  making 

1  =  0  in  equation  (232)  and  finding  the  value  of  t. 


Therefore, 


base  =  \/3 , 
Base  =  2  1/3. 


168  CIRCUITS  CONTAINING 

The  height  is  found  by  making  t  =  0  and  finding  the  value 
of  i. 

Therefore,     Height  =  unity. 

(234)  Hence  Area  parabola  =  f  [base  X  height]  =  J  1/3. 

The  area  of  the  sine-curve  is  equal  to  the  mean  ordinate 

multiplied  by  the  base  ;  therefore 

• 

Area  sine-curve  =  mean  ordinate  X  n. 

The  mean  ordinate  of  a  sine-  curve  equals  twice  the  maxi- 
mum ordinate  divided  by  n.  [See  p.  37.]  By  equation  (233), 
the  maximum  ordinate  equals  -f  1/3  and,  therefore,  mean 
ordinate  =  f  n  V3,  and 

(235)  Area  sine-curve  =  ^  1/3, 

which  is  the  same  as  the  area  of  the  parabola  given  in  (234) 
above.  Moreover,  the  tangents  of  the  angles  which  these 
two  curves  make  at  the  point  0  with  the  axis  are  equal, 
and  the  curves  consequently  blend  into  one  another  with- 
out any  abrupt  change  in  continuity.  This  is  easily  shown 
as  follows  :  Differentiating  (232)  and  (233)  respectively,  we 
have 

(236)  -^7=  --t«  =  taii0. 


(237)  =  $  f3  cos  t  =  tan  8'. 

C/i    V 

Making  t  =  1/3  in  (236),  we  have  the  tangent  of  the  inclina- 
tion of  the  parabola  at  the  point  0.  Making  t  =  —  n  in 
(237),  we  have  the  tangent  of  the  inclination  of  the  sine- 
curve  at  the  point  0.  These  values,  it  is  noticed,  reduce 
(236)  and  (237),  respectively,  to  tan  8  =  tan  6'  =  -  f  1/3, 
which  is  the  value  of  the  tangent  of  inclination  of  either 
curve  at  the  point  0. 


RESISTANCE,  SELF  INDUCTION,  AND  CAPACITY.      169 

It  remains  to  find  the  /  i*  d  t  for  each  of  these  curves. 
By  transposition,  the  equation  of  the  parabola  (232)  is 


••='-!• 


By  squaring, 


Integrating  between  the  limits  —  4/3  and  V3,  we  have 


4  4/38      2  I/36      16 

4 


This  is  the  /  i*dt  effect  for  the  parabola. 
For  the  sine-curve  the  equation  is 
i  =     V3  sin  t. 


r^dt  =|  Adn2£. 

Integrating  between  the  limits  0  and 

/*-.  4 

^o    ^^^X 


This  gives  the  /V  c^  t  effect  for  the  sine-curve.     Hence  we 

find  that,  although  the  area  of  the  current  curve  is  the  same 
for  the  positive  and  the  negative  current  —  that  is,  the  total 

algebraic  quantity  of  flow  is  zero  —  yet  the  I  i*dt  effect  is 


170  CIRCUITS  CONTAINING 

different  in  the  positive  and  negative   directions.     In  the 
case  supposed,  the  ratio  of  the  two  effects  is 

-fe  =  1.135. 


This  may  afford  an  explanation  for  the  fact  that  in  many 
cases  one  carbon  of  an  alternating-current  arc  lamp  is  con- 
sumed more  rapidly  than  the  other,  depending  upon  the 
way  it  is  connected  up. 

General  Proof.  —  Let  us  now  return  to  the  consideration 
of  the  energy  in  a  conductor  when  any  periodic  E.  M.  F.  is 
applied,  and  ascertain  whether  there  is  any  condition  by 
which  self-induction  and  capacity  may  be  introduced  into 

the  circuit  without  changing  the  energy  or  /  i*  d  t  effect. 
The  energy  expended  in  a  conductor  is  proportional  to 
R     When  the  E.  M.  F.  in  the  circuit  is 


e  =  '^sin  (bcot+  0),     [see  (221),] 

E,b,B 

which  represents  any  periodic  E.  M.  F.,  it  has  been  shown 
that  the  current  is 


(238)      i  = 


co 


neglecting  the  complementary  function  [see  (223)].  And, 
when  there  is  neither  self-induction  nor  capacity,  the  cur- 
rent is 

(239)  i0  =    >  -     sin  (&  «>  t  +  B). 


•.-;-; 

RESISTANCE,  SELF  INDUCTION,  AND  CAPACITY.      Ill 
If  we  put 
(240)  1  = 


we  may  abbreviate  (238)  and  (239)  as  follows  : 
(242)  i  =  ^>  /  sin  (6  GO  t  +  a). 


(243)  iQ  =.-          70  sin  (6  &  *  +  0). 

The  subscript  0  indicates  the  absence  of  self-induction 
and  capacity.  Kemembering  that  the  energy  is  proportional 
to  I  ?dt,  we  have 

(244)  W=  yV  d  t  =,/  [^  /  sin  (6  at  t+  <f)Td  *, 
and 

(245)  JF0  =  y  t0a  dt=f\  ^>10  sin  (5  w  <  +  0)1  *  d  «, 

where  TF  is  proportional  to  the  energy  expended  in  the 
circuit  with  L  and  C,  and  W0  bears  the  same  relation  to 
the  energy  when  they  are  absent.  In  order  to  find  wha^ 
relation  must  exist  between  L  and  C  to  cause  the  energy 
expended  during  a  certain  time  to  be  the  same  in  both 
cases,  we  must  integrate  (244)  and  (245)  between  the  same 
limits  of  time,  and  equate  them.  In  order  to  simplify  (244) 
and  (245),  express  as  follows: 


(246)  r^ 

(247)  TF0=y[//sin(&l0^+^+^ 


172  CIRCUITS  CONTAINING 

Since  the  square  of  any  polynomial  is  equal  to  the  sum  of 
the  squares  of  each  term  separately  plus  twice  the  product 
of  each  term  by  every  other  term,  we  have  as  a  result  to 
find  the  integrals  of  only  two  forms,  thus  : 

(248)  f  sin'  (6  GO  t  +  a)  d  t ,     and 

(249)  f  sin  (6,  GO  t  +  a,)  sin  (&,<»  t  +  a,).Jlt 

If  the  limits  are  taken  from  t  =  0  to  t  =  T,  a  complete 

27T 

period, — the  E.  M.  F.  being  periodic  with  a  period  T  =  - 

GO    * 

— it  can  be  shown  that  all  the  integrals  of  the  form  of  (249) 
vanish ;  for,  expressing  the  sine  of  the  sum  of  two  angles 
in  terms  of  the  sines  of  the  angles  themselves, 

(250)  sin(&,G?£  +  ai)  —  s^n  bt&t  cos  cxl  -|-  sin  al  cos  boot, 
and 

(251)  sin(52ft?  t  -f-  <*2)  =  sin  62  GD  t  cos  ora  +  s^n  ^2  cos  ^  °°  ^ 

Multiplying  (250)  and  (251),  we  obtain  terms  of  the  follow- 
ing forms : 

(252)  /  sin  b1  GO  t  cos  b^cotdt, 

(253)  J  cos  b^tcosb^cjtdt 

(254)  /  sin  bl  GO  t  sin  b^aotdt, 

which  are  to  be  integrated  between  the  limits  0  and  T,  or 
— .  Substituting  for  &,&?£,  ax,  and  for  b»(&t,  bx,  we  have 

GO 

made  the  integral  in  (249)  depend  upon  the  three  forms, 

(255)  /     sin  a x  cos  bxdx, 


-  j 

RESISTANCE,  SELF  INDUCTION,  AND  CAPACITY.      173 
(256)  /     cos  a  x  cos  b  x  d  x, 

/n 
sin  a  x  sin  bxdx. 
„ 

To  show  that  each  of  these  three  forms  vanishes  between 
the  limits  zero  and  2  TT,  we  can  reduce  as  follows : 


(258)  r    sin  ax  cos  bxdx  =  <t  /*    sin(a  -\-b)xdx  -I-  ~  /* 

«/o  *t/«  ^t/o 

1  2;r  l~cos(a  -f-  6)0;      cos  (a  —  b)x~i 
a    9  \ a  —T~  o  a  —  u 

(259)  C    cosaxcosbxdx  =  ^  C    cos(a -\- b)x d <v -}- -=  C 

1  27r  rsin(a  —  b)x      sin(a  4-  b)~~\ 
cos(«  -.b)xdx  =  2    o|-^-J--+  -^JJ  =  0. 

/>27r  1        /»27r  1        />27r 

(260)  /     sin  axsinbxdx  =  ^   /     cos(a  —  b)xdx  —  ~   / 

t/o  ^  e/o  ^  t/o 


Since,  therefore,  the  integral  in  (249)  is  zero  in  every  case, 
we  have  only  to  find  the  integral  expressed  in  (248).    This  is 

2w  2_»r 

r*  T  boot  4- a 

(261)    J      sina(&  GO  t  -f  a)d  t  =  ^ 


—  TT — sin  2  (b  cot  -4-  a)     =  —  =  -^< 
±bco  co       2 


which  is  obtained  by  the  formula 


x      1 
/'sin2  x  d  x  =  ~  —  ^  sin  2#, 


174  CIRCUITS  CONTAINING 

upon  replacing  x  by  b  GO  t  -f-  of,  and  dx  by  boodi.  Returning 
to  equations  (246)  and  (247),  and  replacing  the  value  of  the 
integral  in  (261),  as  determined,  we  have  now  found  the 
values  of  W  and  TF0  in  equations  (246)  and  (247)  to  be 


and 


e"°  +/.""  +  etc.]  |, 


Equating  TFand  TF0,  as  before  explained,  to  determine  the 
condition  necessary  to  make  the  energy  the  same,  we  obtain 


(262) 

which,  written  in  full,  is 


[See  (240)  and  (241).]  This  equation  expresses  the  relation 
which  must  be  true  if  the  i  i*dl  effect  is  the  same  when 

the  Bell-induction  and  capacity  are  present  as  it  is  when 
they  are  absent.  This  equation  expressed  without  the  sign 
of  summation  is 

E*  E* 

(264)  ' 


E*      E* 

+  etc.  =  -—  +  ^-  +  etc. 


RESISTANCE,  SELF  INDUCTION,   AND   CAPACITY.      175 

It  is  evident  that  the  parenthesis  in  the  denominator  of  each 
term  of  the  first  member,  being  squared,  is  always  positive 
no  matter  what  values  Z  and  C  may  have.  Each  term, 
then,  of  the  first  member  is  less  than  the  corresponding 
term  in  the  second  member,  unless  the  expression  in  the 
parenthesis  is  zero.  And  in  order  that  the  first  member 
shall  be  as  large  as  the  second  member,  each  parenthesis 
must  be  separately  equal  to  zero ;  that  is,  we  must  have 


and  &,&?  =  — ,     etc. 

VLU* 

Therefore  b1  =  b^  =  bs  —  etc.  But  this  condition  is  equiv- 
alent to  saying  that  the  impressed  E.  M.  F.  can  only  be  a 
simple  harmonic  E.  M.  F.,  and  that  we  must  have  the  rela 

tion  GO  =  — in  order  to  have  the  Ctfdt  effect  the  same 

in  a  circuit  when  the  self-induction  and  capacity  are  pres- 
ent as  when  they  are  absent.  There  is,  then,  no  relation 
between  the  self-induction  and  capacity  which  can  be  given 

that  will  make  the  I  i*  dt  effect  the  samte  in  a  circuit  when 

they  are  present  as  when  they  are  absent,  if  the  impressed 
E.  M.  F.  is  not  an  harmonic  E.  M.  F. 


CHAPTER  XII. 

CIRCUITS  CONTAINING  DISTRIBUTED  CAPACITY  AND  SELF 
INDUCTION.     GENERAL  SOLUTION.* 

CONTENTS:— Derivation  of  the  differential  equations  for  circuits  containing 
distributed  capacity  only.  This  equation  extended  so  as  to  represent 
a  particular  case  of  distributed  capacity  and  self-induction.  Differ 
ential  equation  for  E.  M  F.  is  of  the  same  form  as  that  for  current. 
The  general  solutions  of  the  differential  equations.  Particular 
assumption  of  harmonic  E.  M.  F.  Constants  of  the  general  equation 
determined  under  this  assumption;  first,  from  the  exponential  solu- 
tion; second,  from  the  sine  solution.  Current  determined  from  the 
E.  M.  F.  equation. 

IN  former  chapters  the  only  capacity  considered  has 
been  that  due  to  a  condenser  placed  at  some  particular 
point  of  the  circuit,  thus  introducing  an  actual  break  in  the 
continuity  of  the  conducting  metal.  It  is  possible  to  have 
the  effects  of  capacity  without  thus  introducing  a  condenser 
into  the  circuit.  The  problem  of  the  propagation  of  the 
electric  current  in  a  cable  containing  distributed  static 
capacity  was  first  discussed  by  Sir  William  Thomson,  and 

*  The  purpose  in  writing  this  book  has  been  to  give  concisely  such 
principles  as  are  necessary  for  a  clear  understanding  of  alternate-current 
phenomena,  and  to  make  the  work  one  connected  unit,  dealing  with  the 
various  problems  in  turn,  so  that  no  portion  could  be  omitted  without  in- 
terfering with  the  logical  sequence.  This  and  the  following  chapter  con- 
stitute, however,  a  separate  discussion  which  may  be  read  alone,  and 
\vithout  which  the  rest  of  the  book  is  logically  complete. 

176 


DISTRIBUTED   CAPACITY  AND  SELF  INDUCTION.     Ill 

afterwards  by  Mascart  and  Joubert,*  Blakesley,  f  and 
others.  The  solution  for  the  variation  in  the  current  and 
potential  at  different  points  of  a  conductor  containing  self- 
induction  as  well  as  distributed  capacity  was  given  by  the 
authors  in  the  American  Journal  of  Science  %  and  some  of 
the  effects  of  the  self-induction  noted,  and  a  fuller  dis- 
cussion was  given  in  the  London  Electrician.^ 

When  a  current  of  electricity  flows  in  a  wire,  the  po- 
tential of  the  wire  at  any  point  is  generally  different  from 
the  potential  of  the  surrounding  medium,  and  in  order 
that  this  potential  may  be  different  it  is  necessary  that  the 
exterior  surface  of  the  wire  should  become  charged  with  a 
certain  amount  of  electricity.  A  portion  of  the  current, 
then,  as  it  flows  along  the  wire,  is  used  to  charge  the  sur- 
face of  the  wire.  Indeed,  the  wire  must  be  charged  with 
its  proper  amount  before  the  current  can  flow  on  to  more 
distant  parts  of  the  circuit.  It  is  evident,  then,  that  the 
larger  the  capacity  of  the  wire  to  hold  a  charge,  the  greater 
will  be  its  effect  in  modifying  the  flow  of  current.  The 
capacity  per  unit  length  of  the  wire  (the  wire  being  re- 
garded as  one  plate  of  the  condenser)  depends  upon  its 
superficial  area  and  upon  the  thickness  of  the  dielectric 
(usually  between  it  and  the  conducting  earth  near  it),  as 
well  as  upon  its  nature.  In  Fig.  43  is  represented  the 
longitudinal  section  of  a  cable,  A  being  the  conducting  wire 
and  BB  the  insulating  sheath  around  it.  Suppose  it  to 
be  submersed  in  water ;  the  other  conductor  is  the  water, 
which,  with  the  wire,  forms  the  condenser. 

Let  the  capacity  of  a  unit  length  of  the  wire  be  denoted 
by  (7,  and  the  capacity  of  an  element  PQ,  whose  length  is 

*  Mascart  and  Joubert,  Le9ons  sur  1'electricite  et  le  magnetisme.  Vol.  I., 
§233. 

f  T.  H.  Blakesley,  Alternating  Currents  of  Electricity,  Cliap.  VIII. 

t  Vol.  XLIV.,  page  389. 

§  Vol.  XXIX.,  pages  619  and  634. 


178  CIRCUITS  CONTAINING 

d  x,  by  Cd  x.  Let  R  denote  the  resistance  of  a  unit  length 
of  the  wire.  The  resistance  of  the  element  PQ  is  Rdx. 
Suppose  a  current  i  is  flowing  across  the  section  of  the 
wire  at  P  in  the  positive  direction  indicated  by  the  arrow. 
Let  the  potential  of  P  at  that  instant  be  denoted  by  e. 


FIG.  43.  —  LONGITUDINAL  SECTION  OP  CABLE. 

Since  the  current  always  flows  from  the  higher  to  the  lower 
potential,  the  potential  at  Q,  the  other  end  of  the  element, 
must  be  less  than  that  at  P,  and  the  potential  therefore 
diminishes  in  the  positive  direction.  This  fall  of  potential 

de 

from  P  to  Q  is  denoted  by  —  -j—dx.     By  Ohm's  law  the 

d  x 

current  i,  at  any  moment  through  the  element  PQ,  equals 
the  difference  of  potential  divided  by  the  resistance,  and 
is,  therefore, 

de  _ 

j—dx  .,    -, 

•  **  I  a  e 


If  the  current  remained  constant,  having  this  value  i  all  the 
time,  the  potential  of  the  element  and  its  charge  would 
continually  remain  the  same,  and  the  flow  of  electricity 
across  the  section  Q  would  be  the  same  as  that  at  P,  since 
as  much  must  flow  out  from  as  into  the  element,  unless  the 
charge  of  the  element  be  changed.  Now,  considering  that 
the  current  does  not  remain  constant  but  changes  every 
moment  of  time,  the  potential  e  of  the  element,  and  conse- 
quently its  charge,  must  change  with  the  time.  When  the 
charge  changes,  it  means  that  more  electricity  is  flowing 


DISTRIBUTED   CAPACITY  AND  SELF  INDUCTION.     179 

into  than  out  from  the  element,  or  vice  versa,  and  conse- 
quently the  flow  of  current  across  P  is  different  from  that 
.across  Q  by  just  such  an  amount  as  the  element  gains  or 

loses.     The  current  at  Q  is  then  denoted  by  i  -f  ^—  dx. 

d  x 

Let  the  quantity  flowing  across  the  section  P,  in  the 
time  d  t,  be  denoted  by  d  Q,  and  that  across  the  section  Q 
by  d  Q  —  d  q,  where  d  q  is  the  change  in  the  charge  of  the 
element  in  the  time  d  t.  The  quantity  of  electricity  flowing 
across  the  section  P  is  equal  to  the  current  flowing  at  P 
multiplied  by  the  time  ;  that  is, 

(266)  dQ  =  idt,    or    |y  =  £ 

Similarly  the  flow  across  Q  is  the  current  flowing  at  Q 
multiplied  by  the  time  ;  that  is, 

(267)  d  Q  -  dq  =  (i  +  d^d  x]d  t. 
Subtracting  (267)  from  (266),  we  obtain 


This  equation  may  be  interpreted  to  mean  that  the  rate  of 
change  of  the  charge  on  the  element  is  equal  to  the  differ- 
ence of  the  currents  flowing  into  the  element  and  out  from 
it.  We  might  at  once  have  written  this  equation  from  this 
•consideration. 

The  charge  of  the  element,  as  of  any  condenser,  is  equal 
to  its  capacity  multiplied  by  its  potential.  The  charge 
being  denoted  by  q,  the  potential  by  e,  and  the  capacity,  as 
stated  above,  by  Cd  x,  we  have 

(269)  '  q—  Cedx. 


180  CIRCUITS  CONTAINING 

The  rate  of  change  of  the  charge  with  the  time  is,  by  dif 
ferentiation, 


Equating  this  result  to  equation  (268),  we  have 


Equations  (265)  and  (271)  are  the  differential  equations 
which  are  sufficient  to  determine  the  problem  of  the  propa- 
gation cf  the  current  along  a  cable  containing  distributed 
capacity  such  as  that  described,  when  the  impressed 
E.  M.  F.  of  the  source  is  known.  The  solution  of  these 
equations  may  be  obtained  for  the  most  general  case,  al- 
though the  arbitrary  constants  of  integration  can  only  be 
determined  in  certain  particular  cases  where  the  impressed 
E.  M.  F.  is  known. 

When  the  impressed  E.  M.  F.  is  harmonic  and  equal  to 
e  =  E  sin  a>  t,  the  arbitrary  constant  may  be  found. 

These  two  differential  equations  may  be  expressed  as  a 
single  equation  by  differentiating  (265)  with  respect  to  x 
and  equating  to  (271),  thus  : 

di  _         l<Fe 
dx~    ~ 


(272)  and,  therefore,          =GR~t' 

In  the  foregoing  discussion  no  account  has  been  taken 
of  the  self-induction  of  the  circuit,  but  it  necessarily  has  a 
certain  effect  upon  the  flow  of  the  current  which  it  would 
be  well,  if  possible,  to  consider.  The  effect  of  the  self- 
induction  must  be  felt  as  a  back  E.  M.  F.  opposing  the 
current  and  depending  upon  its  rate  of  change.  We  shall 
assume  that  the  back  E.  M.  F.  per  unit  length  of  the  con- 


DISTRIBUTED  CAPACITY  AND  SELF  INDUCTION.  181 
ductor  is  equal  to  the  rate  of  change  of  the  current,  multi- 
plied by  a  constant ;  that  is,  it  is  equal  to  L  -r-'  In  some 

cases  this  assumption  may  approximately  represent  the 
true  effect  of  self-induction,  and  it  is  thought  that  this  par- 
ticular assumption  may  show  the  nature  of  the  effect  of 
self-induction  even  in  cases  where  the  assumption  is  not 
justifiable. 

Instead  of  leaving  equation  (265)  as  it  stands,  therefore, 
without  taking  into  account  the  effect  of  the  back  E.  M.  F. 
of  self-induction,  we  may  introduce  this  effect  into  the 
equation  by  subtracting  from  the  difference  of  potential 

de 
between  P  and  Q,  viz.,  j-  d  xt   the   internal  E.  M.  F.   of 

di 

self-induction,  L  TT  d  x,  and  so  may  write,  still  in  accord- 
ance with  Ohm's  law, 

(de  .          r  di  -    \ 

(-j—dx  —  L-j-.dx)  *     -,         T    ,. 

-       ^dx  dt        -  -  !^-e  +  ^^« 

Edx  Rdx^  R  dt 

The  relation  in  equation  (271)  is  not  changed  by  the  con- 
sideration of  the  self-induction,  and  these  two  equations, 
(271)  arid  (273),  are  sufficient  to  determine  the  problem  of 
the  flow  of  current,  taking  into  account  both  the  capacity 
and  self-induction.  These  equations,  now  containing  four 
variables,  may  be  expressed  as  two  differential  equations 
containing  three  variables  by  eliminating  first  i  and  then  e. 
After  transposing  and  arranging,  we  may  write  (271)  and 
(273) 


182  CIRCUITS  CONTAINING 

Operating  upon  (274)  by  -j-,  that  is,  differentiating  with 

d  t 

respect  to  t,  we  obtain 


(276)  C,    +_^  =  0. 


Operating  upon  (275)  by  j-  -3—  ,  we  find 


Adding  (276)  and  (277),  we  have 

~tfe      1  d*e      Rdi 


Substituting   here   the   value    of  ^,  namely,  —  C  ^  ,  in 

(274).  we  eliminate  t  and  finally  have  for  the  differential 
equation  of  potential 


To  eliminate  e  from  (274)  and  (275),  operate  upon  (274) 
by  -T-,  and  upon  (275)  by  C-r-.,  and  we  have 


(281)        and      0  --  -  L  C^  +  £0%  =  0. 


f  I 

DISTRIBUTED  CAPACITY  AND  SELF  INDUCTION.    183 

Subtracting  (281)  from  (280),  we  have  the  differential 
-equation  for  current 


It  is  evident  from  the  similarity  of  equations  (279)  and 
^282)  that  the  integral  current  equation  will  be  the  same 
as  the  integral  potential  equation,  except  for  the  arbitrary 
-constants  that  enter  in  integration. 

To  FIND  THE  SOLUTIONS  OF  THE  DIFFERENTIAL  EQUATIONS. 

Assume  that  the  solutions  of  the  pair  of  differential 
equations  (274)  and  (275)  are 

mx  +  nt 

<283)  e  =  Jc  e 

mx  +  n  t 

<284)  and    i  =  e 

where  m,  n,  and  k  are  constants  which  must  be  determined, 
and  x  is  the  distance  from  the  source  of  E.  M.  F.  These 
•constants  may  be  determined  by  differentiation  so  that  the 
-equations  satisfy  the  differential  equations  (274)  and  (275), 
and  are,  therefore,  correct  solutions.  Differentiating  (283) 
and  (284)  with  regard  to  x  and  t,  we  obtain 

d  6  ,      mx+nt 

..    .1       yyy)      L*    s~  • 

—     // 1    /I/    C  • 

dx 

mx+nt 


di  mx  +  nt 

dx 


—  =  me  , 


-r  dl         T        mx  +  nt 

L  -=-  =  Lne 
at 


184  CIRCUITS  CONTAINING 

Substituting  these  values  in  (274)  and  (275),  we  obtain  the- 
simultaneous  equations 

(285)  nkC+m  =  0, 

(286)  and    mk-Ln  +  fi  =  0. 

If  these  equations  are  satisfied,  the  differential  equations 
are  likewise  satisfied.     Solving  for  m  and  n,  we  find 

CkR 

(287)  m= 


(288)  +c¥+2- 

Substituting  these  constants  in  (283)  and  (284),  we  have 

(289)  e  =  Jc 


B      (t-Ckx) 


r-(t-Ckx) 


(290)  and    i  =  ec] 

These  equations  are  solutions  of  equations  (274)  and' 
(275),  and  they  may  be  easily  verified  by  differentiation. 
But  a  more  general  solution  might  be  obtained  by  assum- 
ing the  E.  M.  R,  e,  to  be  a  sum  of  several  terms  such  as 
that  already  assumed,  thus  : 


mt+nt  mx+nt  ^^  ,  7     mx+nt 

(291)  e  =  \\e          +  h,  Jc,  e  +  .  .  .  =  ^^  tike 

h,k. 


/onm    •         -L       mx+nt    ,     7       mx  +  nt  ^^  ,      mx+nt 

(292)  ^=:&1e  +^3e  -)-...  =  ^^  h  e 


DISTRIBUTED  CAPACITY  AND  SELF  INDUCTION.    185 

Determining  m  and  n  as  before,  (291)  and  (292)  may  be  ex- 
pressed 

(293)         /  e= 

ft,*, 


(294)  /     * 

ft,  fc. 

These  equations  may  also  be  verified  by  differentiation 
and  found  to  satisfy  the  differential  equations  (274)  and  (275), 
and  they  are  the  complete  integrals  of  those  differential 
equations.  If  we  know  how  the  current  or  the  potential 
varies  with  the  time  at  any  one  point  of  the  wire,  the  arbi- 
trary constants  h  and  k  can  be  determined,  and  we  have 
the  complete  solution  of  the  problem,  and  are  enabled  to 
tell  the  potential  or  current  at  every  point  of  the  wire  at 
any  time. 

HARMONIC  E.  M.  F. 

The  general  solutions,  (293)  and  (294),  hold  true  in  case 
the  constants  to  be  determined  are  real  or  imaginary  ;  if 
they  are  imaginary,  the  equations  may  be  transformed  into 
a  real  form  consisting  of  some  function  of  the  sine. 

Suppose  the  cable  before  described  is  indefinitely  long, 
and  that  at  the  point  P  (arbitrarily  selected  as  the  zero 
point  of  the  wire,  the  positive  direction  being  indicated  by 
the  arrow)  the  potential  is  caused  to  vary  harmonically 
with  the  time  and  is  always  equal  to 

(295)  e  =  Esma)t. 

Since  equation  (293)  expresses  the  E.  M.  F.  at  every 
point  of  the  wire  and  at  every  moment  of  time,  we  may,  by 
making  x  =  0  in  that  equation,  find  an  expression  giving 


186  CIRCUITS  CONTAINING 

the  potential  at  the  origin  at  every  moment  of  time.     This 
•expression  is 


(296)  e= 


But,  since  we  have  supposed  this  potential  to  be  harmonic, 
we  may  equate  equation  (295)  to  (296)  and  determine  the 
constants  h  and  Tc  so  as  to  make  the  expressions  identical. 
Equating  the  equations  thus,  we  have 


*       r* 


(297) 

h,k. 

In  order  to  determine  the  constants,  we  write  the  sine  in 
its  exponential  form,  thus  : 


e  —  e 

sm  cot  =  - 


. 

where  j  stands  for  V  —  1.  (See  equation  (109),  Chapter  VII., 
with  footnote.)  We  may,  therefore,  substitute  in  (297)  the 
exponential  value  of  the  sine  and  write  only  two  terms  of  the 
summation,  thus  : 

(298)  e 


This  becomes  an  identity  if 

(299)  '    and     h>k>  =  - 


R 

(300)    Also,  if  J<x>=-*r>    and    -y 


DISTRIBUTED   CAPACITY  AND  SELF  INDUCTION.    187 

(3  00} 
Solving  equations  (38)  for  &,  and  &2 ,  we  have 


(301) 


(302)  k,=  ± 

VC&> 


Since  all  the  constants  are  found  to  be  imaginary,  this 
imaginary  exponential  expression  for  the  E.  M.  F.  can  be 
transformed  into  a  real  expression  involving  some  function 
of  the  sine.  This  sine-function  may  be  found  by  continuing 
the  method  already  indicated.  The  next  step  necessary  is- 
to  transform  the  complex  imaginary  values  of  &  by  a 
rather  laborious  process  until  the  imaginary  j  is  removed 
from  under  the  radical  sign. 

It  will  be  evident  that  the  following  equations  are 
identically  true,  either  by  squaring  .each  member  and  seeing 
that  they  are  identical,  or  by  supposing  either  R  or  L  to 
be  zero,  when  they  reduce  to  an  identity. 


(303) 


(304) 


188 


CIRCUITS  CONTAINING 


Substituting  these  expressions  in  (301)  and  (302),  and  writ- 
ing Im  for  the  impedance  (7?a  -f-  1?  <*rf>  we  have 


*= 


Since  we  know  that 


=:,    and     I/TT=— 


we   may  substitute   these  values   in   (305)  and   (306)  and 
write 


(307)     k,  =  ±  j  —  ;=[ 
(  2  v  C  GO 


+  R] 


(308)    k.=  ±     —7= 

GO 


These  values  of  &x  and  k^  may  be  simplified,  for  we  have 
the  identities 


(309)  Vim  -  E+Vlm  +  R  =  1/2  Vlm  +  Z  GO, 
and 

(310)  Vlm-B  -  Vlm  +  R  =  V2  Vim -Loo. 


DISTRIBUTED  CAPACITY  AND  SELF  INDUCTION.     189 

These  may  be  verified  by  squaring  both  members.  Upon 
the  substitution  of  these  values,  the  expressions  for  \  and 
k  become 


(311)    k,  =  ±  yg  Vlm  +  La>  +j  Vim-Loo}' 


<312)     ^= 


Returning  to  equation  (293)  of  E.  M.  F.'s  and  writing  two 
terms  of  the  summation,  we  have 

Rt  Ck^Rx  Rt  Ck^Rx 

(313)     e  =  h,kleck''  +  L    cv  +  i  +  /ij^eo^+z,-cv  +  r 

T) 

Substituting  in  (313)  the  values  of  ht  ,  \  ,  ht  ,  k^  ,  ni  /,    T, 

L>  Kl    -J-  x/ 

T^ 

and  pjL/i  ^  given  in 


<814)        e  =  ^ 

Substituting  in  (314)  the  values  of  the  constants  k,  and 
/£2 ,  already  given  in  equations  (311)  and  (312),  and  factoring 

out  the  common  factor  e          2  x ,  we  have 

(315)    e  =  Ee± 

(  ~  ~W~  '  ) 


190 


CIRCUITS  CONTAINING 


Eemembering  the  exponential  value  of  the  sine,  equation: 
(109),  we  may  express  (315)  as  a  sine-function,  thus  : 


sin     cot  ± 


(7<» 


m 


L 


This  equation  gives  the  value  of  the  potential  at  any  point 
of  the  conductor  at  any  time.  Its  interpretation  and  dis- 
cussion will  be  taken  up  in  the  following  chapter. 

SECOND  METHOD  OF  OBTAINING  THE  SOLUTION. 

We  might  have  assumed  the  solution  to  be  some  func- 
tion of  the  sine,  since  the  potential  at  the  origin  is  supposed 
to  vary  harmonically  ;  and  it  is  much  easier  to  determine 
the  constants  if  we  do  make  such  an  assumption,  inasmuch 
as  we  need  not  deal  with  imaginaries.  Let  us  assume  that 
the  solution  is  of  the  form 


e  = 

and  determine  the  arbitrary  constants  #,  /?,  h,  r,  and  p  so 
as  to  satisfy  the  differential  equation  (279).  We  see  that 
the  constant  r  must  be  zero,  for  when  x  is  zero  the  E.  M.  F. 
is  E  sin  GO  t.  Therefore  li  —  E,  and  /3  =  GO.  The  constants 
p  and  a  remain  to  be  determined,  and  the  E.  M.  F.  is 

(317)  e  =  Eepx  s 

By  differentiation  we  obtain 


ax)  -\-2pa 


e    cos  (a)t-\-  ax} 


px 


—  CRcoe    cos(c<ot-{-ax)~ 


DISTRIBUTED  CAPACITY  AND  SELF  INDUCTION.    191 

Equating  the  coefficients   of    the    sine    and  cosine,   sepa^ 
rately,  to  zero,  we  obtain  the  simultaneous  equations 

p*  -  a*  -  CL  of  =  0, 
and          2p  a—  C  R  <v  =  0. 
Solving  these  equations  for  p  and  a,  we  find  that 


/Coo 
(319)  and   a  =  ±  A  /  - 

V      2 

Substituting  in  (317)  these  values  of  the  constants  p  and  a, 
we  find  that  the  result  is  identical  with  (316)  already  ob 
tained  by  a  different  method. 

To  OBTAIN  THE  CURRENT. 

The  current  may  be  obtained  from  the  potential  equa- 
tion by  means  of  the  relation   C  _ry  =  —  -r-   [see  (271)]. 

Differentiating  (317)  with  respect  to  t  and  multiplying  by 
(7,  we  obtain 


- 
dt  dx 

Integrating  this  result  with  respect  to  #,  we  get 


i  =  -  —  —  ]  p  cos  (GO  t  -\-  a  x)  -\-  a  sin  (GO  t  -\-  a  x)  j-  . 

Transforming  this  into  an  equation  containing  the   sine 
only,  by  means  of  formula  (27),  Chap.  III.,  we  obtain 


192    DISTRIBUTED   CAPACITY  AND  SELF  INDUCTION. 


VCOO  px          ( 

(320)    i=—E-1-=    --  e     smlGot 
Vtf  +  D  ^  ( 


/  Im  —  L 
-f  tan-'A/- 
V 


Here  p  and  a  represent  the  expressions  (318)  and  (319). 
Equation  (320)  may  be  written 


( 


sin     oot±  ax+  tan  -  1 


This  equation  gives  the  value  of  the  current  at  any  time 
at  any  point  of  a  conductor  containing  distributed  capacity 
and  self-induction  when  subjected  to  an  harmonic  source 
of  electromotive  force.  The  discussion  of  this  equation 
and  the  potential  equation  will  be  taken  up  in  the  follow- 
ing chapter. 


CHAPTER  XIII. 


CIRCUITS  CONTAINING  DISTRIBUTED  CAPACITY  AND 
SELF  INDUCTION.— DISCUSSION. 

•CONTENTS  : — Circuits  icitJi  no  self-induction.  Particular  form  of  e  and  i 
equations.  Nature  of  waves.  Rate  of  propagation.  Wave-length. 
Decreasing  amplitude.  Rate  of  decay  with  distance, — with  time. 

Circuits  with  self-induction.  Phase  difference.  Rate  of  propagation 
Diminishing  amplitude.  Rate  of  decay.  Limitations  of  the  tele- 
phone. 

Wave  propagation  in  Closed  Circuits. 

Positive  and  negative  waves  travel  around  the  circuit  until  they 
vanish.  Resultant  effect.  Potential  zero  at  middle  point  of  the  cable. 
Expression  for  potential  simplified  if  the  length  of  the  cable  is 
a  multiple  of  a  wave-length.  Same  results  may  be  applied  to  the 
current  equation. 

IN  order  to  ascertain  the  physical  effects  of  distributed 
self-induction  and  capacity  in  a  circuit,  we  will  first  discuss 
the  analytical  results  obtained  in  the  preceding  chapter,  as 
applied  to  a  circuit  in  which  the  self-induction  is  neglected, 
and  then,  after  investigating  the  nature  of  the  wave-propaga- 
tion, the  wave-length,  rate  of  propagation,  and  rate  of  decay, 
consider  circuits  containing  distributed  capacity  and  self- 
induction,  noting  the  changes  caused  by  the  introduction  of 

the  self-induction. 

193 


194  CIRCUITS  CONTAINING 


CIRCUITS  WITH  DISTRIBUTED  CAPACITY  BUT  NO  SELF  INDUCTION. 

When  the  effect  of  self-induction  is  not  considered  in 
the  cable,  we  may  put  L  =  0  in  the  current  and  potential 
equations  (316)  and  (320)  and  reduce  them  to  more  simple 
forms,  thus  : 


(321)       e  = 


C(a 

(.322)       t= 


sin 


These  results  agree  with  those  given  by  Mr.  T.  H.  Blakesley 
in  his  book  on  "  Alternating  Currents  of  Electricity,"  page 
60,  second  edition.  They  may  be  directly  obtained  from 
the  differential  equation  (272),  and  upon  differentiation  will 
be  found  to  satisfy  it. 

Nature  of  the  Wave-propagation. — Equation  (321)  shows 
that  at  any  point  of  the  conductor  the  potential  varies  har- 
monically with  the  time .  At  the  origin  where  x  =  0,  the 
potential  is  always  equal  to  e  =  ^sin  GO  t,  its  maximum  value 
being  E\  but  as  we  proceed  from  the  origin  the  potential 

T  --/<D5* 

becomes  less,  being  equal  to  E  e  2  ,  an  expression 
which  decreases  as  x  increases.  The  double  sign  is  retained 
in  the  exponent  in  the  equation,  since  it  represents  two 
waves,  one  going  in  the  positive  and  the  other  in  the 
negative  direction  from  the  origin,  which  is  the  source 
of  alternating  potential.  The  maximum  value  of  the  po- 
tential at  any  point  of  the  cable  may  be  represented 
as  in  Fig.  44  by  an  ordinate  to  the  logarithmic  curve ; 


DISTRIBUTED   CAPACITY  AND  SELF  INDUCTION.     195 

thus,  OA  represents  the  maximum  value  of  the  harmon-, 
ically  varying  potential  at  the  origin,  and  .Z?6yits  maximum 
value  at  a  distance  x  from  the  origin.  The  distance  OB  1S 
taken  as  that  in  which  the  logarithmic  curve  has  decreased 


to  --of  its  original  value,  OA.      The  distance    OD  is   a 
quarter  wave-length,  and  OE  a  half  wave-length.     It  will 


TTiG.  44. — CURVE   I. — INSTANTANEOUS  WAVE  IN  INFINITE  CABLE. 
CURVE  II. — LOGARITHMIC  DECREASE  IN  AMPLITUDE. 


presently  be  shown  that  the  amplitude  decreases  to  almost 
-g-iF  of  its  original  value  in  one  complete  wave-length. 

One  of  the  most  striking  results  shown  by  these  formulae 
is  that  the  current  always  precedes  the  potential  by  one 
eighth  of  a  period,  and  this  difference  in  phase  is  not  altered 
l>y  any  change  in  the  resistance  or  capacity  of  the  cable. 

Bate  of  Propagation. — Curve  I.  (Fig.  44)  represents  an 
instantaneous  position  of  the  potential  wave  travelling  along 
the  wire  in  each  direction  from  the  origin.  The  distance 


196  CIRCUITS  CONTAINING 

along  the  cable  from  one  maximum  potential  to  the  next,  or 
in  other  words  the  length  of  one  complete  wave,  may  be 
found  by  equating  to  2  TT  the  angle  containing  x  in  the 
equation.  This  gives 


X— 


or        x 


/~~2  l~    n 

—  A  —  2  n  A  /  -FTO —  =  2  \  /  ^  p  - , 
y   6  R  GO          y   0  Mn 


where  A  denotes  the  length  of  one  complete  wave,  and 
n  =  ~ —  the  frequency  of  alternation. 

A  7t 

The   time   of   one   complete   period   is  represented  by 

T  =  — ,  and  in  that  time  the  wave  advances  a  distance  A, 
n 

equal  to  one  wave-length.  The  rate  at  which  the  wave 
advances  is  found  by  dividing  the  wave-length  by  the  time 
taken  in  advancing  that  distance  ;  thus, 


A.  /  n  n  1  2  GO 

ion  =  ^r  =-  2  */  -^-g  —  \L  -^-^  . 


Bate  ot  propagation 


It  is  seen  that  the  rate  of  propagation  not  only  depends 
upon  the  character  of  the  cable,  but  likewise  varies  as  the 
square  root  of  the  frequency  of  alternation.  A  wave  with  a 
frequency  of  400  will  travel  twice  as  fast  as  one  with  a 
frequency  of  100  alternations  per  second. 

Decreasing  Amplitude. — The  frequency  affects  not  only 
the  rate  of  propagation,  but  also  has  a  marked  influence 
upon  the  rate  at  which  the  amplitude  of  the  waves  decreases 
with  the  distance.  The  distance,  which  we  will  call  x,  at 

which  the  amplitude  of  the  wave  will  have  -  of  its  original 


DISTRIBUTED  CAPACITY  AND  SELF  INDUCTION.     197 

value,   is   the   reciprocal    of   the   coefficient   of  x  in   the 
exponent,  thus : 


»'  =  -i/_2 


This  distance,  at  which  [the  wave  decays  to  —  of  its  value 

at  the  origin,  varies  inversely  as  the  square  root  of  the 
frequency  ;  this  means  that  a  wave  with  a  frequency  of  100 
alternations  per  second  can  go  twice  as  far  as  one  with  a 
frequency  of  400  and  experience  the  same  decay  in  ampli- 
tude. Comparing  this  value  of  x'  with  that  of  a  wave-length 
A,  we  may  write 


=  97- 

The  amplitude  of  a  wave,  therefore,  decreases  to  -^  =  .00187 

of  its  value  in  advancing  a  distance  equal  to  one  wave- 
length. 

In  order  to  find  the  time,  t',  in  which  the  amplitude  de- 
cays to  -  of  its  value,  we  must  divide  this  distance,  a?',  by 
the  rate  of  propagation,  thus  : 


_ 

2  7T 


We  see  that  the  time  of  decay  varies  inversely  as  the 
frequency  ;  thus  a  wave  with  a  frequency  of  400  alternations 
per  second  will  decrease  a  certain  amount  in  one  fourth 


198  CIRCUITS  CONTAINING 

the  time  a  wave  with  a  frequency  of  100  is  decreasing  the 
same  amount. 


CIRCUITS  WITH  DISTRIBUTED  CAPACITY  AND  SELF  INDUCTION. 

It  will  be  noticed  that  the  introduction  of  self-induction 
into  the  cable,  in  the  manner  before  described,  does  not 
materially  alter  the  character  of  the  wave  propagation. 
This  is  evident  from  the  equations  (316)  and  (320).  At 
every  point  of  the  conductor  the  potential,  or  current,  varies 
harmonically  with  the  time,  and,  as  before,  the  amplitude  of 
the  wave  decreases  with  a  logarithmic  decrement  as  it  pro- 
ceeds from  the  origin. 

Phase  Difference. — One  effect  of  the  self-induction  is  to 
change  the  angle  of  advance  of  the  current  ahead  of  the 
potential.  This  angular  difference  is  no  longer  a  constant 
angle  of  45°,  as  formerly,  but  is  a  function  depending  upon 
R,  L,  and  GO,  viz., 


(323)  tan0  =  A/^-4^  =  -- 

V  Im  +  Lw      a 


When  L  =  0,  tan  6  —  1,  and  6  —  45°.  As  L  increases  from 
zero  to  infinity  the  expression  changes  from  unity  to  zero, 
and  the  angle  0,  consequently,  from  45°  to  0°.  The  effect 
of  the  self-induction  is,  therefore,  to  decrease  the  phase 
difference  between  the  current  and  potential.  This  difference 
of  phase  also  becomes  less  with  an  increase  in  the  frequency. 
Rate  of  Propagation. — The  distance  along  the  cable 
between  two  maxima  is  found  by  equating  the  angle  con- 
taining x  in  equation  (316)  to  2  n  and  solving  for  x,  thus  : 


2  TT  2  n  1/2 

x  =  A.  =  -  -  =  — 

OC  A/  /IM  A/  7  ™ 


DISTRIBUTED   CAPACITY  AND  SELF  INDUCTION.     199 

where  Ax  denotes  the  wave-length.      Subscripts  are  here 
used  to  denote  circuits  with  self-induction.     The  time  oc- 

2?r 
<mpied  in  travelling  this  distance  is  T  = ;  hence, 


A,  GO  I 

Bate  of  propagation  =  -^  =  —  =  A  /  -^rj 


It  is  to  be  noticed  that  the  wave-length  and  the  rate  of 
propagation  are  each  less  than  that  found  for  circuits  con- 
taining no  self-induction.  When  L  =  0,  these  expressions 
just  found  reduce  to  those  previously  given. 

Another  point  to  be  noticed  is  that  a  change  in  frequency 
will  have  a  greater  effect  in  altering  the  wave-length  and 
not  so  great  an  effect  in  changing  the  rate  of  propagation 
as  in  the  case  of  a  circuit  with  no  self-induction.  Two 
waves  of  different  periods  will,  therefore,  go  more  slowly 
but  with  less  difference  in  their  rates  of  propagation  than 
with  no  self-induction.  As  before,  the  wave  of  higher  fre- 
quency will  have  the  shorter  wave-length  and  advance  the 
faster. 

Decreasing  Amplitude. — As  before,  the  amplitude  of  the 
harmonic  wave  has  a  logarithmic  decrement,  decreasing 
with  the  distance  from  the  origin.  The  distance  at  which 

the  amplitude  has  —  of  its  original  value  is  the  coefficient 
of  the  exponent,  thus  : 


P       Vim  —  L  OD 

This  is  larger  on  account  of  the  self-induction.  The 
substitution  of  L  —  0  reduces  it  to  the  value  of  of-  found 
before.  An  increase  in  frequency  will  cause  this  value  to 


200  CIRCUITS  CONTAINING 

decrease,  and  the  decay  in  a  certain  distance  to  increase  ; 
but  the  frequency  has  not  so  great  an  influence  upon  this 
decay  as  it  has  in  a  circuit  with  no  self-induction. 

Comparing  with  the  values  found  for  A:  and  #,  we  see 
that 


"  2  TT  tan  0  * 

In  order  to  find  the  time,  £/,  in  which  the  amplitude 
decays  to  —  of  its  original  value,  we  divide  the  distance 
a?,'  by  the  rate  of  propagation,  thus  : 

\ 

,  _  2  TT  tan  8  _        T  I 

A.          ~  2  n  tan  6  ~  GO  tan  0* 


The  self-induction  causes  tan  6  to  have  a  value  less  than 
unity,  thus  increasing  the  time  for  a  certain  decay, — that  isr 
decreasing  the  rate.  An  increase  in  the  frequency  causes 
6  to  become  smaller.  The  exact  effect  upon  the  rate  of 
decay  caused  by  a  variation  in  frequency  in  a  circuit  with 
self-induction  depends  upon  the  constants  of  the  circuit. 
The  wave  of  higher  frequency  will  always  decay  the  more 
rapidly,  but  with  self-induction  in  the  circuit  there  is  less 
difference  in  the  rates  of  decay  of  waves  of  different  periods, 
than  there  is  in  circuits  without  it. 

Limitations  of  the  Telephone. — The  effects  of  distributed 
capacity  in  a  conductor  upon  the  wave-propagation  have 
been  given,  and  the  way  in  which  these  effects  are  altered 
by  the  introduction  of  self-induction  or  by  a  change  in 
frequency.  A  consideration  of  the  results  with  reference 
to  telephone  circuits  is  valuable  inasmuch  as  it  is  just  such 
effects  that  cause  the  limitations  to  telephony.  In  all  cases; 


• 
DISTRIBUTED  CAPACITY  AND  SELF  INDUCTION     201 

the  waves  of  higher  frequency  travel  the  faster,  and  so  the 
several  harmonic  components  of  a  complex  tone  are  con- 
stantly changing  in  their  relative  phases.  The  waves  of 
higher  frequency  are  likewise  subject  to  the  more  rapid 
decay,  and  so  when  the  several  components  are  recombined 
the  resultant  tone  may  be  materially  altered  from  the  orig- 
inal complex  tone.  These  effects  may  be  modified  by  the 
presence  of  self-induction,  but  in  all  cases  they  will  be 
present  to  a  certain  extent,  thus  defining  the  limits  of  the 
use  of  the  telephone. 

WAVE-PROPAGATION  IN  CLOSED  CIRCUITS. 

Let  us  consider  that  a  dynamo  giving  an  harmonic  alter- 
nating E.  M.  F.  is  inserted  at  some  point  of  a  cable,  such 
as  that  described,  which  forms  a  continuous  closed  circuit. 
There  will  be  a  forward  wave  of  positive  potential  starting 
from  one  pole  of  the  machine  which  will  travel  around  and 
around  the  circuit,  continually  diminishing  in  amplitude, 
until  it  finally  vanishes.  At  the  same  time  a  backward 
wave  of  negative  potential  will  start  from  the  other  pole  of 
the  machine  and  travel  around  in  the  opposite  direction 
until  it  too  vanishes.  The  potential  at  any  particular  point 
of  the  circuit  is  thus  the  sum  of  the  potentials  due  to  all 
the  positive  and  negative  waves.  Let  I  denote  the  length 
of  the  cable.  When  the  first  positive  wave  reaches  a  point 
at  a  distance  x  from  the  pole  of  the  dynamo,  the  potential 
due  to  this  forward  wave  is 


e  F  =  E  e   j     sin  (GO  t  —  a  x), 

where  p  and  a  have  the  values  given  in  (318)  and  (319). 
When  the  wave  has  travelled  completely  around  the  circuit 
and  comes  to  this  point  a  second  time,  its  value  is 

—  a  x  —  a  I), 


202  CIRCUITS  CONTAINING 

which  may  be  obtained  by  substituting  x  4-  I  for  x  in  the 
above.     After  going  n  times  around,  it  has  become 


-  a  a?  - 


The  resultant  of  all  these  forward  waves  at  any  one  point 
may  be  written,  therefore, 


n=  oo 


(324)    ep=eFi 


sin  (oot  —  ax  —  aln). 


When  we  consider  all  the  backward  waves,  they  may  be 
represented  by  a  similar  expression  in  which  E  has  the 
negative  sign.  When  the  first  backward  wave  has  travelled 
around  the  circuit  so  as  to  be  a  positive  distance  x  from  the 
origin,  it  has  travelled  a  distance  I  —  x ;  this  distance  from 
the  origin  in  a  negative  direction  we  will  call  x'.  We  may 
therefore  write  for  the  first  backward  wave 

and  for  the  sum  of  all  the  backward  waves 

tt,  =  oo 

(325)  eB  =  —  Ee~px'  ^>e~pln  sin  (GO t  —  a  x'  —  a  I n). 

n  =  0 

These  expressions  (324)  and  (325)  may  be  simplified 
since  we  may  put 

(326)  sin  (cot  —  ax  —  aln)  =  sin  (GO  t  —  a  x)  cos  aln 

—  cos  (GO  t  —  a  x)  sin  aln. 


DISTRIBUTED  CAPACITY  AND  SELF  INDUCTION.     203 
Substituting  (326)  in  (324),  we  have 


(327)     e=£e-pxsw(&t-  ax)  ^e~pln  cos  aln 


—  Ee~px  cos(cot  —  ax)  ^e~pln  sin  aln. 

n=0 

Similarly  we  may  reduce  (325)  to 

(328)     eB  =  -  Ee~px'  sin  (cot  -  ax')  ^>e~pln  cos  aln 

11=00 

p*'  cos((*)t-ax')^>e-plnsinaln. 


The  resulting  potential  at  any  point  due  to  the  forward 
and  backward  waves  is  the  sum  of  EF  and  EB.  Writing 
I  -  x  for  x'  in  (328)  and  adding  to  (327),  we  obtain 

(329)     e  =  E^*e-plncosaln  \  e~px  sin  (GO  t  -  ax) 

n  =n  ( 


ax  —  al)  f.  -f  £^>e~pln  sin  aln 

71  =  0 


By  means  of  the  exponential  values  of  the  sine  and  cosine, 
the  values  of  the  two  summations  expressed  in  (329)  are 
found*  to  be 


-. 
^6          sin  aln  =  — 


*  Equations  (330)  and  (331)  may  be  verified  thus  :  For  brevity  put 
pi  —  h,  and  a  I  .=  k.  Writing  the  exponential  value  of  the  sine  [see  equa- 
tion (109),  Chap.  VII  ],  we  have 


204  CIRCUITS  CONTAINING 

and 

%T  -Pin  e**>l-e*    cosal 

(331)         ;>e          co$aln  =  -  -      pl        -—  - 

— 


cos 


n  =  0 

n  =00 


syf 

Jkn - hn 


- 


Thus  we  have  the  given  series  equivalent  to  the  difference  of  two  infinite 
decreasing  geometrical  series.  The  sum  of  such  a  series  is  known  to  be 
equal  to  frhe  first  term  divided  by  unity  minus  the  common  ratio,  i.e., 

s  — ,  where  s  denotes  the  sum,  a  the  first  term,  and  r  the  common 

ratio.    Applying  this  formula,  the  sum  of  the  first  series  is  — -  • -je-h ' 

and  of  the  second  -  .  • ____  .     Hence 

n  =  oo  / 

"5T-   _-*n__.  1  1  1 


~(jk 


Multiplying  both  numerator  and  denominator  by  e  h  aud  reducing  the  terms 
in  the  brackets  to  a  common  denominator  after  factoring  out  the  factor  e  ft, 
we  have 


Replacing  the  exponential  values  of  the  sine  and  cosine,  we  have 


In  a  similar  manner  we  may  verify  equation  (331),  thus 


DISTRIBUTED   CAPACITY  AND  SELF  INDUCTION.     205 

Let  POP  (Fig.  45)  represent  the  cable  which  is  sup- 
posed to  form  a  closed  circuit,  the  ends  at  P  being  joined 


FIG.  45. — FORWARD  AND  BACKWARD  WAVES,  AND  RESULTANT  POTEN- 
TIAL, IN  A  CLOSED  CONDUCTOR. 

together.  The  maximum  value  of  the  potential  at  the  posi- 
tive pole  of  the  dynamo  is  represented  by  O  A.  As  we  go 
from  A,  this  decreases  along  the  logarithmic  curve 
A  B  CD  E,  until  it  finally  vanishes  altogether.  Similarly,  a 


Therefore 


n  =  0 


cosal-\- 


Q.E.D. 


206  CIRCUITS  CONTAINING 

backward  wave  coming  from  the  negative  pole  decreases 
along  the  curve  A'  B'  C'  D'  E' .  At  the  point  P,  half-way 
between  the  poles  of  the  dynamo,  the  middle  point  of  the 
cable,  it  is  evident  that  the  potential  remains  continually 

I 

zero,  for,  at  the  point  P,  the  distance  x  is  ~- ,  which  reduces- 

equation  (329)  to  zero. 

If  the  length  of  the  cable  happens  to  be  some  multiple 
of  a  wave-length,  the  expression  for  the  potential  takes  a 
simpler  form.  In  this  case  each  successive  forward  wave 
travels  around  the  circuit  in  the  same  phase  as  the  first, 
and  all  these  forward  waves  may,  therefore,  be  added 
together  algebraically.  The  maximum  resultant  potential 
at  any  point  will  be  the  sum  of  the  maxima  of  the  separate 
waves. 

In  Fig.  45,  eF  ,  ep  represent  successive  forward  waves, 
and  e  and  e  the  corresponding  backward  waves.  In  the 
case  where  the  length  of  the  cable  is  a  multiple  of  the  wave- 
length, the  sum  of  the  maxima  of  all  the  forward  and  of  all 
the  backward  waves  is  represented  by  the  dotted  lines  eF 
and  eB,  respectively.  The  solid  line  e,  the  sum  of  ep  and 
eB,  represents  the  resultant  maximum  potential  along  the 
conductor. 

27T 

We  have  seen  that  the  wave-length  is  A  =  --  .    The 

a 

length    of   the   cable    is   a   multiple    of   the  wave-length 

I  —  K\  =  ,  and  al  —  2  n  x,  where  K  is  a  constant 

a 

This  value  reduces  (330)  to  zero,  since  sin  2  K  it  =  0,  and 
reduces  (331)  to 


DISTRIBUTED   CAPACITY  AND  SELF  INDUCTION.    207 

These  values  cause  the  second  term  in  (329)  to  vanish,  and 
the  whole  becomes 


(332) 


px  _ 


which  expresses  the  resultant  potential,  represented  by  the 
solid  line  e  in  Fig.  45,  at  any  point  of  the  cable,  provided  its 
length  is  some  multiple  of  a  wave-length.  When  x  =  0,  this 
reduces  to  e  =  E  sin  oo  t,  the  expression  for  the  potential 

at  the    terminals  of  the  dynamo.     When  x  =  ^  ,  the  ex- 

A 

pression  vanishes,  showing  that  the  potential  is  constantly 
zero  at  the  middle  point  of  the  conductor. 

This  last  simplification  was  made  possible  by  consider- 
ing the  length  of  the  cable  to  be  a  multiple  of  the  wave- 
length ;  otherwise  the  algebraic  addition  of  the  maxima  of 
the  several  waves  would  not  be  possible,  since  they  would 
differ  in  phase.  The  construction  of  the  resultant  curves 
would  not  be  so  simple,  but  the  nature  of  the  results  would 
not  be  materially  modified. 

The  phenomena  in  connection  with  the  flow  of  current 
are  similar  to  those  just  discussed  relating  to  the  propaga- 
tion of  potential,  and  are  obtained  in  the  same  manner  from 
the  current  equation. 


PART  II. 

GRAPHICAL   TREATMENT. 


CHAPTER  XIV. 

INTRODUCTORY  TO  PART  II.  AND  TO  CIRCUITS  CON- 
TAINING RESISTANCE  AND  SELF  INDUCTION. 

CONTENTS: — Introductory.  Analytical  solutions  of  Part  I.  for  simple 
circuits  extended  to  compound  circuits  by  graphical  method.  Arrange- 
ment of  Part  II.  Graphical  representation  of  simple  harmonic 
E.  M.  F.'s.  Graphical  representation  of  the  sum  of  simple  harmonic 
E.  M.  F.'s  of  same  period.  Triangle  of  E.  M.  F.'s  for  a  single  circuit 
containing  resistance  and  self-induction.  Impressed  E.  M.  F.  Ef 
fectiveE.  M.  F.  Counter  E.  M.  F.  of  self-induction.  Direction  shown 
from  differential  equation.  Graphical  representation.  Methods  to  be 
used  and  symbols  adopted  in  the  graphical  treatment  of  problems. 
First  method  (the  one  used  throughout  this  book),  employing  E.  M.  F. 
necessary  to  overcome  self-induction.  Second  method,  employing 
E.  M.  F.  of  self-induction.  System  of  lettering  and  conventions 
adopted  in  graphical  construction. 

THE  analytical  solutions  derived  in  Part  I.  apply  merely 
to  a  single  circuit  having  resistance,  self-induction,  and  ca- 
pacity in  series.  The  problems  which  arise  in  case  there 
is  not  simply  a  single  circuit  but  a  complicated  network  of 
conductors  might  be  treated  analytically,  though  the  pro- 
cess would  be  exceedingly  laborious  and  the  results  too 
cumbersome  to  handle.  Fortunately,  however,  by  making 
use  of  the  analytical  solutions  already  given  in  Part  I.,  and 
extending  them  by  graphical  methods,  we  are  enabled  to 
solve  problems  with  compound  circuits  which  offer  con- 

211 


212  CIRCUITS  CONTAINING 

siderable  difficulty  to  analytical  investigation.  These 
graphical  methods  are  most  easily  and  advantageously 
adapted  to  problems  in  which  we  deal  with  an  harmonic 
impressed  E.  M.  F.  :. 

The  object  of  this  Part  is  to  show  how  to  solve  by 
graphical  methods  any  problems  arising  with  any  combina- 
tion of  series  and  parallel  circuits,  in  any  branch  of  which 
there  may  be  an  harmonic  impressed  E.  M.  F. 

The  plan  to  be  followed  is  similar  to  that  adopted  in 
the  first  Part.  First  are  considered  various  compound  cir- 
cuits which  contain  resistance  and  self-induction  only,  and 
then  circuits  containing  resistance  and  capacity  only,  and 
finally  circuits  containing  all  three,  resistance,  self-induc- 
tion, and  capacity.  The  problems  to  be  considered  in  each 
case  are  similar,  first  a  series  circuit,  then  a  divided  circuit 
with  two  branches  and  with  any  number  of  branches,  then 
any  combination  of  series  and  parallel  circuits. 

Before  giving  the  solutions  of  these  problems,  the  way 
in  which  this  graphical  method  corresponds  to  and  is  a 
substitute  for  the  analytical  method,  and  the  manner  in 
which  it  is  to  be  used,  will  be  explained. 

GEAPHICAL  [REPRESENTATION  OF  A  SIMPLE  HARMONIC 
ELECTROMOTIVE  FORCE. 

An  harmonic  impressed  electromotive  force  is  repre- 
sented by  the  equation 

e  =  E  sin  GO  t, 

as  was  explained  in  Chap.  II.  on  harmonic  functions.  The 
plot  of  this  equation,  in  which  t  is  the  independent  and  e 
the  dependent  variable,  gives  the  sine-curve  represented  in 
Fig.  46.  A  diagrammatic  method  of  representing  this  har- 
monic E.  M.  F.  is  seen  in  the  same  figure.  The  line  OA 
is  supposed  to  revolve  in  the  counter-clockwise  direction 


RESISTANCE  AND  SELF  INDUCTION. 


213 


about  the  point  0  with  uniform  angular  velocity.  Its  pro- 
jection 0  P  at  any  moment  corresponds  to  the  ordinate 
O'P'  of  the  sine-curve.  If  the  circle  be  moved  horizontally 
with  a  constant  velocity,  the  projection  0  P  would  trace  a 
sine-curve  the  ordinates  of  which  represent  the  value  of  the 
impressed  E.  M.  F.  at  any  instant.  Diagrammatically  we 
may  represent  the  impressed  E.  M.  F.  by  the  line  0  A 


FIG.  46.  —  GRAPHICAL  REPRESENTATION  OP  A  SIMPLE  HARMONIC 
ELECTROMOTIVE  FORCE. 

alone,  which  is  equal  in  length  to  its  maximum  value,  E. 
In  this  sense,  then,  we  may  represent  harmonic  E.  M.  F.'s 
by  lines  in  the  graphical  constructions  which  follow. 


GRAPHICAL  KEPRESENTATION  OF  THE  SUM  OP  SIMPLE  HAR- 
MONIC ELECTROMOTIVE  FORCES  HAVING  THE  SAME  PERIOD. 

If  an  E.  Si.  F.  is  the  sum  of  two  simple  harmonic  E.  M.  F.'s 
of  the  same  period,  it  may  be  represented  by  the  equation 


(333) 


e  —  E,  sin  GO  t  +  E^  sin  (GO  t  +  0). 


It  can  easily  be  shown  analytically  that  this  sum  is  a 
simple  harmonic  E.  M.  F.,  differing  in  phase  and  amplitude 
from  each  of  the  two  components,  and  having  the  same 


214 


CIRCUITS  CONTAINING 


period  ;   for,  upon  expanding   sin   (GO  t  -f-  #),  the   equation 
becomes 


e  =.  (E1  -(-  E^  cos  0)  sin 


sin  0  cos  GO  t. 


This  may  be  transformed  by  means  of  the  trigonometric  for- 
mula (27),  Part  L,  to 


(334)  e  = 


*  +  2  E,  E,  cos  0 


sin 


^  sm  6      ) 


I,  COS0  f  * 

This  equation  represents  a  simple  harmonic  E.  M.  F.,  since 
it  is  of  the  form 

e  —  E  sin  (GO  t  -\-  0), 

in  which  j^and  0  are  constant  quantities.  Moreover,  this 
equation  shows  that  the  diagonal  of  the  parallelogram 
formed  by  the  two  component  lines  which  represent  the 
two  component  terms  of  equation  (333)  is  the  line  which 
graphically  represents  equation  (334),  and  is  therefore  the 
sum  of  the  two  components. 


FIG  47. —RESULTANT  OF  Two  HARMONIC  ELECTROMOTIVE  FORCES. 

In  Fig.  47,  curve  I.,  generated  by  the  line  0  A,  represents 
the  first  term  of  equation  (333).  Curve  II.,  generated  by 
OB,  represents  the  second  term.  Curve  III.  is  the  sum  of 


RESISTANCE  AND  SELF  INDUCTION.  215 

curves  I.  and  II.,  and  is  generated  by  the  diagonal  O  (J  of  the 
parallelogram  formed  upon  the  two  components  0  A  and 
0  B.  That  curve  III.,  the  geometrical  sum,  represents 
equation  (334),  the  analytical  sum,  is  seen  by  the  fact  that 
the  analytical  relations,  as  shown  by  the  equation,  agree 
with  those  readily  obtained  from  the  geometry  of  the 
figure.  Thus,  from  the  equation,  the  amplitude  of  the 
resultant  harmonic  function  must  be 


E  =  <JE?  +  E:  +  2  E,  E,  cos  ft 
But  from  the  geometry  of  the  figure  this  same  relation  is 


evident,  for  ~OA  =  E,  ,  0~B  =  E^*n&AOB  =  0. 

Again,  from  the  equation  the  resultant  E.  M.  E.  differs 
in  phase  from  E^  by  an  angle 


From  the  figure  we  see 


nn   A  iGD  i 

0  =  C  0  A  =  tan'1  -=s=-  —  tan"1  -^ 


This  agreement  of  the  analytical  and  graphical  relations 
establishes  the  correctness  of  the  construction,  and  we  can, 
therefore,  conclude  that  the  sum  of  any  two  sine-curves  of 
the  same  period  represented  by  two  lines  revolving  about 
a  common  centre  is  also  a  sine-curve  of  the  same  period 
represented  by  the  diagonal  of  the  parallelogram  formed 
on  the  two  component  lines. 

When  the  component  E.  M.  F.'s  are  more  than  two  in 
number  the  sum  is  represented  by  the  vector,  which  is  the 
geometrical  resultant  of  all  the  component  vectors.  This 
evidently  follows  from  the  preceding,  since  any  two  compo- 


216 


CIRCUITS  CONTAINING 


nents  are  equivalent  to  a  single  E.  M.  F.,  and  this  combined 
with  a  third  and  fourth  component  gives  the  geometrical 
resultant  as  the  sum  of  all  the  components.  Thus,  in  Fig* 


FIGS.  48  AND  49. — ADDITION  OF  HARMONIC  ELECTROMOTIVE  FORCES. 

48,  we  have  a  number  of  vectors  A,  B,  C,  D,  each  repre- 
senting one  component  E.  M.  F.  and  drawn  from  the  same 
origin  0.  The  sum  is  found  in  the  usual  manner  by  con- 
structing a  parallelogram  on  any  two  and  then  combining 
the  resultant  with  a  third,  and  so  on  until  all  the  compo- 
nents are  reduced  to  a  single  resultant  vector  R.  This 
process  is  equivalent  to  that  indicated  in  Fig.  49,  where  the 
vector  A  is  first  drawn  from  the  origin  0,  then  B  from  the 
extremity  of  A,  C  from  the  extremity  of  13,  and  so  on  until 
all  the  lines  are  drawn.  The  resultant  or  geometrical  sum 
is  then  the  vector,  R,  drawn  from  the  origin  to  the  last 
point  found,  thus  completing  a  closed  polygon. 


RESISTANCE  AND  SELF  INDUCTION.  217 

TRIANGLE  OF  ELECTROMOTIVE  FORCES  FOR  A  SINGLE  CIRCUIT 
CONTAINING  RESISTANCE  AND  SELF-INDUCTION. 

In  Chapter  III.,  in  which  circuits  containing  resistance 
and  self-induction  were  analytically  treated,  it  was  shown 
that  if  a  circuit  contains  an  harmonic  impressed  E.  M.  F., 

e  =  ^sin  cat, 
the  value  of  the  current  is  also  harmonic  and  is 

(335)        i  =  —JL—  sin  L  t  -  tan"  —}. 


This  current  equation  was  derived  from  the  differential 
equation  of  electromotive  forces 


in  which  e  is  the  instantaneous  value  of  the  impressed 
E.  M.  F.  of  the  source,  and  E  i  that  part,  usually  called  the 
effective  E.  M.  F.,  necessary  to  overcome  the  ohmic  resist- 

ed i 

ance,  and  L  j-,  ^na^  Par^  necessary  to  overcome  the  counter 
d'  t 

E.  M.  F.  of  self-induction. 

In  Fig.  50  let  the  vector  0  A  represent  the  harmonic 
impressed  E.  M.  F.  Then,  by  equation  (335),  we  know  that 
the  current  is  represented  by  a  vector  0  E  lagging  behind 


GO 


_ 

0  A  by  an  angle  6  whose  tangent  is   -^-. 

The  effective  E.  M.  F,  must  be  represented  by  a  vector 
0  C,  equal  to  El,  in  the  same  direction  as  the  current,  and 
equal  to  the  current  vector,  OB,  multiplied  by  R.  The 

counter  E.  M,  F.  of  self-induction,  L  -      is  at  right  angles 


218 


CIRCUITS  CONTAINING 


to  the  current  and  must  therefore  be  represented  by  the 
vector  C  A  perpendicular  to  0  B.    It  can  be  shown  to  be 


FIG.  50. — TRIANGLE  OF  ELECTROMOTIVE  FORCES.  FIRST  METHOD— THE 
ONE  USED  THROUGHOUT  THIS  BOOK— EMPLOYING  E.  M.  F.  TO  OVER- 
COME SELF-INDUCTION. 

at  right  angles  to  the  current,  as  follows.    .Equation  (335) 
may  be  written  thus  : 


i  —  1  sin  (GO  t  —  6). 


By  differentiation, 


(336) 


di 
Ti 


=  (&!  cos  (GO  t  —  6). 


Multiplying  this  equation  by  L  and  writing  in  terms  of  the 
sine,  we  have 


(337) 


r/?* 

=  Zc» /sin (a>$-  0  +  90°). 


di 


By  this  equation  it  is  seen,  that  the  E.  M.  F.,  L  -T--,  necessary 

to  overcome  that  of  self-induction  is  represented  by  a  vector 
C  A,  whose  length  is  L  GO  I,  ninety  degrees  in  advance  of  the 


RESISTANCE  AND  SELF  INDUCTION.  219 

current.  The  E.  M.  F.  of  self-induction  is  equal  and  oppo- 
site to  that  which  is  necessary  to  overcome  it,  and  is  conse- 
quently ninety  degrees  behind  the  current,  represented  by 
the  vector  A  C. 

That  the  foregoing  construction  represents  the  case  and 
fulfils  the  analytical  conditions  expressed  by  the  current 
equation  (335)  may  be  shown  again  by  a  further  comparison 
of  the  geometrical  with  the  analytical  relations.  Thus  in 
Fig.  50  or  51, 


0  C 

This  is  seen  to  correspond  to  the  angle  of  lag  in  equation 
(335).  Also  the  impressed  E.  M.  F.  0  A,  being  the  hypote- 
nuse of  the  triangle  OA  0,  is  equal  to  the  sum  of  the 
squares  of  the  two  sides,  and  therefore 


that  is,    E=VR*r+I?<*?r  =  lVR*  +  D  of. 
E 


and  I  = 


+  Z2 


This  is  seen  to  correspond  to  the  maximum  value  of  the 
current  given  in  equation  (335). 

METHOD  TO  BE  USED  AND  SYMBOLS  ADOPTED  IN  THE 
GKAPHICAL  TREATMENT  OF  PROBLEMS. 

In  the  graphical  treatment  of  circuits  with  resistance 
and  self-induction  there  are  two  methods,  each  equally 
correct,  for  obtaining  the  same  results  depending  upon 
whether  we  consider  the  E.  M.  F.  of  self-induction  or  the 
equal  and  opposite  E.  M.  F.  necessary  to  overcome  it. 


220 


CIRCUITS  CONTAINING 


The  method  in  which  the  E.  M.  F.  necessary  to  over- 
come the  self-induction  is  used  is  shown  in  Fig.  50  and  has 
been  fully  discussed.  In  this  method  of  construction,  the 
impressed  E.  M.  F.  is  regarded  as  made  up  of  the  sum  of 
two  components,  one  the  effective  E.  M.  F.  in  the  direction 
of  the  current,  and  the  other  that  necessary  to  overcome  self- 
induction  ninety  degrees  ahead  of  the  current. 


FIG.   51.— TRIANGLE  OF  ELECTROMOTIVE  FORCES.    SECOND  METHOD, 
EMPLOYING  E.  M.  F.  OP  SELF-INDUCTION. 


The  method  in  which  the  E.  M.  F.  of  self-induction  is 
used  is  shown  in  Fig.  51.  The  point  of  difference  is  that 
the  line  A  C  represents  the  E.  M.  F.  of  self-induction  in- 
stead of  the  E.  M.  F.  necessary  to  overcome  it,  and  is  ninety 
degrees  behind  the  current  instead  of  ahead  of  it. 

In  this  method  of  drawing,  the  effective  E.  M.  F.  which 
drives  the  current  is  regarded  as  the  resultant  of  the  two 
other  E.  M.  F.'s  in  the  circuit,  viz.,  the  impressed  E.  M.  F. 
and  that  of  self-induction. 

Either  of  these  methods,  if  carried  throughout  the  whole 
drawing,  is  correct  and  finally  brings  the  same  results ;  but 
unless  one  method  is  adopted  and  carried  throughout,  there 
is  apt  to  be  confusion.  Unless  otherwise  stated,  the  method 
here  adopted  is  the  first  one,  namely,  that  which  considers 


RESISTANCE  AND  SELF  INDUCTION.  221 

the  E.  M.  F.  as  that  necessary  to  overcome  self-induction,  as 
illustrated  in  Fig.  50. 

In  order  that  the  diagrams  may  be  readily  understood, 
the  arrangement  and  system  of  lettering  adopted  will  be 
explained.  In  all  cases  the  positive  direction  is  counter- 
clockwise, and  the  diagram  is  supposed  to  revolve  counter- 
clockwise around  the  centre  0.  Lines  will  be  designated 
by  letters  in  small  capitals  placed  at  their  extremities. 
The  letters  therefore  designate  points  and  will  be  used 
alphabetically,  beginning  with  A,  in  the  order  in  which  the 
points  are  determined.  Thus  in  Fig.  54  the  line  OA  is 
first  drawn,  then  the  points  B,  (7,  1),  etc.,  are  determined, 
and  the  lines  0  B,  0  C,  B  C>  CD,  etc.,  drawn  in  order. 
All  revolve  counter-clockwise  about  0. 

The  direction  of  lines  representing  E.  M.  F.  or  current 
will  be  indicated  by  arrows,  and,  where  possible,  these 
arrows  will  be  placed  so  as  to  show  where  the  lines  ter- 
minate. In  order  that  lines  representing  current  and 
E.  M.  F.  may  be  distinguished,  the  arrows  for  current  will 
have  a  closed  head,  as  in  the  case  of  the  line  0  A,  Fig.  54, 
and  the  arrows  for  E.  M.  F.  will  have  an  open  head,  as  on 
the  Hue  B  C.  Dotted  lines  are  needed  only  for  the  con- 
struction of  the  figure  or  to  make  clear  some  point  that 
would  otherwise  be  ambiguous  or  doubtful.  When  a 
number  of  lines  terminate  at  one  point  and  are  each  di- 
rected toward  the  point,  it  has  been  found  convenient  to 
avoid  the  confusion  of  the  many  arrows  coming  thus  to- 
gether by  omitting  the  arrows  and  drawing  a  small  circle 
at  the  point,  as  at  6r,  Fig.  54. 


CHAPTER  XV. 

PROBLEMS  WITH  CIRCUITS  CONTAINING  RESISTANCE  AND 

SELF  INDUCTION.     SERIES  CIRCUITS  AND  DIVIDED 

CIRCUITS. 

Prob.       I.  Effects  of  the  Variation  of  the  Constants  R  and  L  in  a  Series 

Circuit.     R  varied.     L  varied. 
Prob.      II.  Series  Circuit.     Current  given. 
Prob.    III.  Series  Circuit.     Impressed  E.  M.  F.  given. 
Prob.  Ilia.  Measurements  on  a  Series  Circuit. 
Prob.    IV.  Divided  Circuit.     Two  Branches.     Impressed  E.  M.  F.  given. 

Equivalent  Resistance  and  Self-induction  defined. 
Prob.      V.  Divided    Circuit.      Any  Number    of    Branches.      Impressed 

E.  M.  F.   given.     Equivalent  Resistance  and  Self-induction 

obtained  for  Parallel  Circuits. 
Prob.    VI.  Divided    Circuit.     Current    given.      First    Method:    Entirely 

Graphical.     Second    Method:    Solution    by  Equivalent  R 

and  L. 
Prob.  VII.  Effects  of  the  Variation  of  the  Constants  R  and  L  in  a  Divided 

Circuit  of  Two  Branches.     R  varied.     L  varied.     Limiting 

Cases.     Constant    Potential    Example.     Constant   Current 

Example. 

Problem  I.    Effects   of  the  Variation   of  the  Constants 
It  and  L  in  a   Series  Circuit. 

BEFOBE  taking  up  the  problems  proper  which  arise  in 
connection  with  the  investigation  of  circuits  containing 
resistance  and  self-induction,  it  will  be  well  to  first  con- 
sider the  changes  which  occur  when  the  resistance  is  varied 
and  the  coefficient  of  self-induction  kept  constant,  and  those 

222 


r  '  i      '*" 

RESISTANCE  AND  SELF  INDUCTION.  223 

which  occur  when  the  self-induction  is  varied  and  the 
resistance  kept  constant.  The  limiting  cases,  when  the 
resistance  or  the  self-induction  approaches  zero  or  infinity, 
will  be  shown,  so  that  the  following  problems  may  be 
applied  to  such  limiting  cases  without  the  confusion  which 
might  otherwise  arise. 


KESISTANCE  VAKIED. 

Let  us  suppose  that  the  ohmic  resistance  is  varied  in  a 
circuit  in  which  the  self-induction  is  kept  constant. 

Let  OACy  Fig.  52,  represent  the  triangle  of  E.  M.  F.'s 


FIG.  52. — VARIATION  OF  RESISTANCE  AND  SELF-INDUCTION  IN  A  SERIES 
CIRCUIT.     PROBLEM  I. 

for  the  circuit  when  the  resistance  is  It.  Divide  0  C  by  R 
to  obtain  the  current  /  equal  to  0  13.  Draw  the  line  O  D 
of  indefinite  length  perpendicular  to  the  E.  M.  F.  0  A  in 
the  direction  of  lag.  The  angle  D  0  C,  being, the  comple- 

Draw  B  E  perpendicu- 


ment  olAOC,  is  therefore  tan"  l— — , 

L  GO 


lar  to  0  B  and  let  it  meet  the  line  O  D  in  E.     Then  in  the 
right   triangle  O  B  E  the  side  B  E  equals  -=  —  ;  for  0  B 

-Lt  GO 


equals  7,  and  tan  E  0  B  equals 


73 

~=  —  . 
-Lt  co 


l}24  CIRCUITS  CONTAINING 

It  follows  that  the  hypotenuse,  0  E,  of  this  triangle  is 

Tjl 

equal  to  y — ,  and  is,  therefore,  a  constant  entirely  inde- 
pendent of  any  variation  in  the  current  /,  or  resistance  R. 
Taking  the  square  root  of  the  sum  of  the  squares  of  the 
sides  0  B  and  B  E,  we  obtain 


From  equation  (29)  we  have 
E 


/=  - 

VR*  +  L1  a? 

Therefore 


OE= 


Now,  since  the  side  OB  of  the  right  triangle  QBE 
always  represents  the  current  /,  and  the  hypotenuse  O  E  is 
independent  of  the  current  or  the  resistance,  it  follows  that 
the  current  is  always  represented  by  a  vector  0  B  inscribed 
in  the  semi-circle  QBE,  for  any  possible  variation  in  the 
resistance.  The  arrow  shows  the  direction  of  change  as  R 
increases. 

In  the  particular  cases  when  R  is  infinite  or  zero  we 
see  clearly  by  this  figure  the  limiting  values  of  the  current. 
When  R  is  infinite  the  current  is  evidently  zero.  When 
R  approaches  zero  (or,  what  is  approximately  the  same 
thing,  becomes  very  small  compared  with  the  self-induc- 
tion) O  B  approaches  0  E,  and  in  the  limit  the  current 
becomes 


-. 

L  GO 


RESISTANCE  AND  SELF  INDUCTION.  225 

When  the  circuit  contains  no  ohmic  resistance  we  see, 
first,  that  C  A  =  0  A,  that  is,  the  impressed  E.  M.  F.  is 
equal  to  L  GO  /,  the  E.  M.  F.  of  self-induction ;  and,  second, 
that  the  current  lags  90°  behind  the  impressed  E.  M.  F. 
These  relations,  here  geometrically  shown,  are  analytically 
expressed  in  equation  (337). 


SELF  INDUCTION  VARIED. 

Suppose  the  coefficient  of  self-induction  is  varied  in  a 
circuit  in  which  the  resistance  is  constant  ;  we  wish  to  find 
how  the  current  changes. 

In  the  same  figure,  52,  prolong  the  line  EB  to  F  until 
it  meets  the  impressed  E.  M.  F.  0  A  prolonged.  Then 

the  line  B  F  must  equal  ~  75™,  since  tan  BOF  equals  -~. 
The  hypotenuse  O  I1  is,  therefore, 

OF  = 


But  from  (29)  we  find 


Do?       E 


Therefore,         OF=  -p. 

Since  the  hypotenuse  0  F  is  independent  of  the  current 
/  or  the  self-induction  Z,  and  is  a  constant  for  any  variation 
in  L,  it  follows  that  the  current  is  always  represented  by  a 
vector,  0  B,  inscribed  in  the  semi-circle  O  B  F,  for  any 
possible  variation  in  the  self-induction  Z.  In  the  figure 
the  arrow  shows  the  direction  of  change  as  Z  increases. 

We  easily  see  what  the  value  of  the  current  is  in  the 


226  CIRCUITS  CONTAINING 

limiting  cases  where  L  is  infinite  and  zero.  When  L  ap- 
proaches infinity,  the  current  approaches  zero.  When  L 
approaches  zero,  the  vector  0  B  approaches  O  f\  the 
E.  M.  F.  necessary  to  overcome  self-induction  is  zero, 

E 

and  the  current  follows  Ohm's  law,  being  equal  to  -T> 

That  the  construction  of  Fig.  52  is  consistent  with  the 
equations  is  further  shown  by  the  following  relations. 


(338) 


(339) 


Equating  (338)  and  (339),  we  find 


a  result  which  is  identical  with  that  analytically  expressed 
in  equation  (335). 

It  is  seen  that  in  the  limiting  cases,  where  the  resist- 
ance or  the  self-induction  approaches  zero  or  infinity,  the 
triangle  of  electromotive  forces  becomes  two  superimposed 
straight  lines,  that  is,  one  side  becomes  zero.  In  most  of 
the  following  problems  only  the  general  cases  are  discussed 
in  which  the  circuit  contains  a  finite  resistance  and  self- 
induction.  The  constructions  may  be  modified,  however, 
according  to  the  principles  just  set  forth,  so  that  the  solu- 
tions given  may  be  applied  to  the  limiting  cases  referred 
to.  Although  in  some  cases  it  may  require  a  little  thought 
and  care  to  make  this  modification,  it  has  been  deemed 
unnecessary  to  show  its  application  to  each  particular- 
problem. 


RESISTANCE  AND  SELF  INDUCTION. 


227 


Problem  II.    Series  Circuit.    Current  Given. 

Let  there  be  a  circuit,  Fig.  53,  having  n  different  coils 


I     ^  Rl,  «-» 


FIG.  53. — PROBLEM  II.  AND  PROBLEM  III. 

in  series,  with  resistances  Rl ,  R^ ,  etc.,  and  self-inductions 
Ll ,  Z2 ,  etc.     It  is  required  to  find  the  impressed  E.  M.  F. 
necessary  to  cause  a  current  /  to  flow  through  the  coils. 
In   Fig.   54   make   OA   equal   to  the  current  flowing. 


FIG.  54. — PROBLEM  II.  AND  PROBLEM  III. 

Multiply  this  by  El  and  lay  off  0  B  equal  RJ,  which  is  then 
the  effective  E.  M.  F.  in  the  first  coil.  Draw  B  C  perpen- 
dicular to  ~O  A  in  the  positive  direction,  or  direction  of  ad- 

7- 

vance,  and  make  the  angle  B  0  C  equal  to  Bl  =  tan'1  -15—. 

Then  B  0  C  is  the  triangle  of  E.  M.  F.'s  for  coil  one,  and 
Ea  is  its  impressed  E.  M.  F.  Similarly  lay  off  CD  parallel 
to  0  A  and  equal  to  7?a/,  and  then  make  the  angle  D  C  E 


228  CIRCUITS  CONTAINING 

equal  to  #„  =  tan"1   -^— .     This  triangle  C D E  then  repre- 
*»« 

sents  the  triangle  of  E.  M.  F.'s  for  the  second  coil,  and  Eb 
its  impressed  E.  M.  F.  In  a  similar  way  we  may  go  on 
constructing  triangles  of  E.  M.  F.'s  for  each  of  the  n  coils 
until  we  finally  reach  a  point  G,  which  is  the  end  of  the  line 
representing  the  impressed  E.  M.  F.  in  the  last  coil.  If  we 
draw  the  line  0  G,  it  must  be  the  impressed  E.  M.  F.  of 
the  source,  which  we  wished  to  find,  as  it  is  the  sum  of  all 
the  n  different  falls  in  potential  for  each  coil.  Indeed,  this 
will  be  evident  from  the  following.  If  we  lay  off  B  H 
=  (77),  and  HK=T?F,  we  find  that  0~K  =  B.I+  R,I 
-f  etc,  —  12  R.  And,  similarly,  JTG  =  Ll  GO  7-f  Z2  GO  I 
-f-  etc.  =  (&I2  L.  If  we  replace  all  the  n  different  coils  by 
a  single  coil  whose  resistance  is  the  sum  of  all  the  n  resist- 
ances, viz.,  2  R,  and  whose  coefficient  of  self-induction  is 
the  sum  of  all  the  n  coefficients,  viz.,  2  Z,  we  find  that  0  G 
is  the  impressed  E.  M.  F.  necessary  to  cause  the  given 
current  7  to  flow,  and  0  K  G  is  the  triangle  of  E.  M.  F.'s 
for  the  equivalent  coil. 


Problem  III.    Series  Circuit.    Impressed  E.  M.  F.  Given. 

First  Method.  —  The  circuit  being  the  same  as  in  PROB- 
LEM II.,  Fig.  53,  it  is  required  to  find  the  current,  I,  which 
a  given  impressed  E.  M.  F.  will  cause  to  flow. 

We  may  solve  this  problem  by  constructing  upon  the 
given  E.  M:  F.  ~6~G,  Fig.  54,  the  triangle  0  K  G  so  that  the 

angle  at  0  is  tan"1  -rcr-     ^ne  s^e  0  K  i^,  then  equal  to 


R.     The  current  /,  —  0  A,  is  then  found  by  dividing 


The  impressed  E.  M.  F.'s,  Ea,  Eb,  etc.,  of  the  several 
parts  of  the  circuit  are  found  as  in  the  previous  problem. 


RESISTANCE  AND  SELF  INDUCTION.  229 


The  total  effective  E.  M.  F.  represented  by  0  K  is  divided 
in  proportion  to  the  resistances  Rl ,  R^ ,  etc.,  into  the  parts 
OB,  B H,  etc.,  representing  the  effective  E.  M.  F.  in  the 
several  parts  of  the  circuit.  The  impressed  E.  M.  F.'s  are 
obtained  by  erecting  upon  0 I>,  B  //,  etc.,  the  E.  M.  F. 
triangles  0  B  C,  C  D  E,  etc. 

Second  Method. — It  is  sometimes  more  convenient  to 
solve  this  problem  in  the  following  way.  Assume  that  a 
certain  current  is  flowing  in  the  circuit,  then  find  the 
E.  M.  F.  required,  by  the  method  of  Problem  II.  Now  if 
the  whole  figure  be  magnified  or  diminished  in  proportion 
until  the  E.  M.  F.  thus  found  is  made  equal  to  the  given 
E.  M.  F.,  then  the  current  will  be  that  due  to  this  given 
E.  M  F.,  which  is  the  required  current ;  for,  it  is  evident 
that  if  we  change  either  the  E.  M.  F.  or  the  current,  the 
other  is  changed  in  proportion,  and  indeed  the  whole  dia- 
gram is  changed  in  proportion. 

Problem  Ilia.— Measurements  on  a  Series  Circuit. 

One  of  the  simplest  and  also  one  of  the  most  important 
cases  of  series  circuits  which  is  met  with  is  that  of  a  non- 
inductive  resistance  in  series  with  an  inductive  resistance 
as  illustrated  in  Fig.  54«.  The  corresponding  diagram  of 
E.  M.  F.'s  is  given  in  Fig.  546,  in  which  OB  and  B  A  rep- 
resent El  and  E^,  the  E.  M.  F.'s  impressed  upon  the  non- 
inductive  and  inductive  resistances,  respectively,  and  O  A 
represents  E,  the  total  impressed  E.  M.  F.  The  inductive 
circuit  7?2  Z2  may  be  the  primary  of  a  transformer  or  any 
inductive  circuit  whatsoever.  From  the  values  of  E>  E^ 
and  _Z£,  which  are  readily  obtained  from  three  voltmeter 
readings,  and  the  value  of  the  non-inductive  resistance  72, , 
we  can  ascertain  the  following  quantities  :  the  angle  6  by 
which  the  current  lags  behind  the  impressed  E.  M.  F.,  E\ 
the  angle  0a  by  which  the  current  in  the  inductive  part  of 


230 


CIRCUITS  CONTAINING 


the  circuit  lags  behind  the  E.  M.  F.,  E^ ,  impressed  upon  that 
part ;  the  impedance,  resistance,  and  self-induction  of  the 
inductive  circuit  (in  the  case  of  a  transformer  it  is  the 
apparent  resistance  and  self-induction  which  is  found) ;  and 


Ef  RJ 

FIGS.  54«,  545. 


R2l  C 


the  power  expended  in  each  part  of  the  circuit  and  in  the 
whole  circuit. 

From  the  values  E,  E, ,  and  E^ ,  the  triangle  0  A  B  is 
drawn,  and  upon  O  A  the  right  triangle  0  CA  is  erected  by 
producing  0  B  to  C. 

The  resistance  J?2  is  obtained  thus.  OB  =  El  7,  B  C 
=  E,  L  Therefore,  E, :  R,  : :  03  :  B~C.  For  R,  we  may 

Tjl 

write      .   The  resistance  Z?2  is  then 


OB    I  ' 


The  angle  6  by  which  the  current  lags  behind  the 
E.  M.  F.  impressed  upon  the  whole  circuit  is  found  from 
the  geometry  of  the  figure.  By  trigonometry, 

E;  =  E*  +£*  -  2  EE,  cos  0. 


Whence        cos  0  = 


RESISTANCE  AND  SELF  INDUCTION.  231 

\—  Jf  'a         W"1 


^ 
i  lL 


The  angle  #a  ,  by  which  the  current  lags  behind  the 
E.  M.  F.  impressed  upon  the  inductive  part  of  the  circuit  is 
similarly  found  from  the  trigonometrical  expression 

E*  =  ^  +  E?  -  2  E,  E,  cos  DBA. 
From  this  it  follows  that 

2    _      7^2    _        ff-t 


cos 


=  —  cos  0  B  A  = 


In  the  non-inductive  portion  of  the  circuit,  the  current 
is  in  phase  with  the  E.  M.  F.  and  0,  =  0. 

The  value  of  the  self-induction  of  the  inductive  circuit 
is  obtained  from  the  values  for  7?a  and  #a  given  above, 

and  from  the  relation  tan  #a  =  —  ^  —  •     The  value   of  the 

-ti2 

expression  Z2  ca,  called  the  inductive  resistance  in  contra- 
distinction to  ohmic  resistance,  may  be  given  in  ohms. 
To  find  Z2,  #2  is  first  found  from  the  expression  given 
above  for  cos  #2  by  means  of  trigonometry  tables,  and 
the  tangent  of  the  angle  is  then  found,  also  from  the  tables, 

and  equated  to  —  ^  —  ,  from  which  Z2  may  be  readily  cal- 

•**! 

culated. 

An  explicit  expression  for  Z3  in  terms  of  E19  E^  and  E 
may  be  found  as  follows.  From  the  figure  it  is  seen  that 


When  the  expression  given  above  for  #a  is  substituted  in 
this  expression,  it  becomes  •  „ 


A  = 


232  CIRCUITS  CONTAINING 

Inasmuch  as  the  expression  involves  the  differences 
of  the  fourth  powers,  it  does  not  afford  as  accurate  a 
method  for  determining  self-induction  as  that  given  in  the 
preceding  paragraph. 

In  these  expressions,  E,  El ,  E^ ,  and  /  represent  maxi- 
mum values,  but  in  the  above  cases  the  expressions  would 
be  the  same  if  the  virtual  values  were  used,  that  is,  the 
square  root  of  the  mean  square  of  the  instantaneous  values 
[see  page  38],  which  are  represented  by  7,  E,  etc.  This  is 
because  the  values  of  the  above  expressions  all  depend 
upon  the  ratio  of  the  quantities  in  such  a  way  that  if  each 
quantity  were  multiplied  by  the  same  constant,  the  values 
of  the  expressions  themselves  would  remain  unchanged. 
It  is  therefore  immaterial  whether  maximum  or  virtual 
values  are  used. 

In  obtaining  the  expressions  for  the  power  expended  in 
each  portion  of  the  circuit  the  virtual  values  will  be  used, 
inasmuch  as  these  are  the  values  usually  obtained  from 
alternating-current  measuring  instruments.  The  general 
expression  for  the  power  expended  in  a  circuit  is  [see  (195) 
and  (196)] 

W=±E  TcosQ  =  JTcos  0, 

where  6  is  the  angle  of  lag  between  the  E.  M.  F.  and  the 
current. 

In  the  non-inductive  resistance  the  angle  of  lag  is  zero 
and  the  power  is,  therefore, 

Wl  =  Ej. 
The  power  expended  in  the  inductive  part  of  the  circuit  is 

W,  =  E,T  cos  B,  =  -£=•  (E*  -~E?-  £.") 

&  -£ 


RESISTANCE  AND  SELF  INDUCTION.  233 

The  power  expended  in  the  whole  circuit  is 

W=  E  /cos  8  =    L  ( £  +  El  -  4°) 


It  is  evident  that  the  power  expended  in  the  whole  circuit 
is  the  sum  of  the  power  in  each  part,  or 

w=  F;+  wz. 

This  method  of  measuring  power  is  known  as  the  three- 
voltmeter  method  and  was  apparently  suggested  by  Mr. 
Swinburne  and  by  Prof.  Ayrton  and  Dr.  Suinpner  at  about 
the  same  time.  The  method  is  applicable  to  any  circuit 
whether  the  E.  M.  F.  is  harmonic  or  not.*  For  maximum 
accuracy  E^  =  E^ . 

Problem  IV.    Divided  Circuit.    Two  Branches. 
Impressed  E.  M.  F.  Given. 

Let  us  consider  the  problem  of  a  divided  circuit  having 
two  branches  in  parallel  as  indicated  in  Fig.  55.  Each 


R2  La 

FIG.  55. — PROBLEM  IV. 

branch  contains  self-induction  and  resistance,  and  there  is 
an  impressed  E.  M.  F.,  E,  between  the  terminals  M  and  N\ 

*  See  "  The  Measurement  of  the  Power  given  by  any  Electric  Current 
to  any  Circuit  :"  Prof.  Ayrton  and  Dr.  Sumpner  ;  Proc.  Boy.  Soc.,  Vol. 
XLIX.,  1891,  p.  424. 


234  CIRCUITS  CONTAINING 

it  is  required  to  fmd  the  main  current,  /,  and  the  currents 
/,  and  J2  in  the  branches. 

Fig.  56  shows  how  to  find  graphically  the  main  and 
branch  currents  when  the  impressed  E.  M.  F.  and  the  resis- 
tance and  self-induction  of  each  branch  are  given. 


FIG.  56.— PROBLEM  IV. 

Since  the  impressed  E.  M.  F.  at  the  terminals  of  each 
branch  is  known,  each  may  be  separately  treated  as  a 
simple  circuit  containing  resistance  and  self-induction  by 
the  method  previously  given  in  Fig.  50. 

Draw  0  A  equal  to  the  impressed  E.  M.  F.,  E.     Make  the 

L    GO 

angle  A  OB  =01=  tan'1  —7^—  in  the  negative  direction  such 

£ll 

that  it  is  an  angle  of  lag.  Then  the  right  triangle  0  B  A 
is  the  triangle  of  E.  M.  F.'s  for  the  first  branch,  0  B  is  the 
E.  M.  F.  necessary  to  overcome  resistance,  and  B  A  that 
necessary  to  overcome  the  self-induction.  In  a  similar  way 

we  may  lay  off  the  angle  A  0  C  =  #2  =  tan'1  -~—  to  repre- 

^i2 

sent  the  angle  of  lag  in  the  second  branch,  and  then 
construct  the  triangle  OCA,  which  will  represent  the 
E.  M.  F.'s  in  the  second  branch.  Since  these  are  right 


RESISTANCE  AND  SELF  INDUCTION.  235 

triangles,  the  points  B  and  C  lie  oh  the  circumference  of  a 
circle  whose  diameter  is  0  A.  Since  the  effective  E.  M.  P., 
Rl  7, ,  in  the  first  branch  is  0  J3,  the  current  is  0  D,  equal 
to  0  B  divided  by  R^  Similarly  the  current  /2  is  0  E, 
equal  to  0  C  divided  by  7?2.  Now  the  current  in  the  main 
circuit  at  any  instant  is  equal  to  the  sum  of  the  currents  in 
the  branch  circuits  at  that  instant.  Construct,  therefore, 
the  parallelogram  upon  the  sides  0  D  and  O  E.  The  diag- 
onal 0  F  represents  the  main  current,  /,  for  its  projection 
at  any  moment  equals  the  sum  of  the  projections  of  the 
two  sides  0  D  and  0  E,  which  projections  represent  the 
instantaneous  values  of  the  current  in  the  two  branches. 
From  the  geometry  of  the  figure  it  follows  that 


It  is  seen  that  the  current  in  each  branch  is  inversely  pro- 
portional to  the  impedance. 

This  diagram  gives  the  complete  solution  of  the  problem 
of  the  divided  circuit.  The  currents  7,  and  72  in  the 
branches  lag  behind  the  impressed  E.  M.  F.,  E\  by  angles 
0,  and  0,.  The  main  current,  7,  lies  between  these,  making 
an  angle  0  with  E.  It  is  evident  that  the  maximum  value 
of  the  main  current,  /,  being  the  longest  diagonal  of  the 
parallelogram  whose  sides  represent  the  currents  in  the 
branches,  is  greater  than  the  current  in  either  branch. 
Since  the  currents  differ  in  phase,  at  certain  parts  of  a 
period  it  happens  that  the  current  in  a  branch  is  greater 
than  the  main  current,  for  when  the  main  current  is  zero 
the  branch  current  may  have  a  considerable  value. 

EQUIVALENT  EESISTANCE  AND  SELF  INDUCTION. 

Suppose  that  instead  of  the  two  parallel  branches  which 
have  been  considered,  a  single  circuit  be  substituted  for  them 
whose  resistance,  ft',  and  self-induction,  Z7,  are  such  that 


236  CIRCUITS  CONTAINING 

the  same  current  as  before  will  flow  in  the  main  line.  Then 
0  G  A  must  represent  the  triangle  of  E.  M.  F.'s  for  this  cir- 
cuit, since  the  impressed  E.  M.  F.  is  O  A,  and  the  effective 
E.  M.  F.  is  0  G,  in  the  direction  of  the  current,  and  the  E.  M.  F. 
G  A  to  overcome  self-induction  is  at  right  angles  to  the 
current.  The  resistance,  R ',  and  self-induction,  L ',  of  the 
equivalent  simple  circuit — that  is,  a  circuit  which  allows  the 
same  current  to  flow  in  the  main  line — are  called  the 
equivalent  resistance  and  equivalent  self-induction  of  the 
divided  circuit. 

The  values  of  this  equivalent  resistance,  R',  and  self- 
induction,  L',  may  easily  be  found  in  terms  of  the  resist- 
ances and  self-inductions  of  the  branches.  This  will  be 
deferred  until  after  the  discussion  of  the  following  problem, 
in  which  the  solution  is  given  for  any  number  of  circuits 
connected  in  parallel. 


Problem  V.    Divided  Circuit.    Any  Number  of  Branches. 
Impressed  E.  M.  F.  Given. 

Let  the  divided  circuit  MN,  Fig.  57,  have  n  branches 


FIG.  57. — PROBLEM  V.  AND  PROBLEM  VI. 

in  parallel,  each  containing  resistance  and  self-induction, 
with  an  impressed  E.  M.  F.,  E,  between  the  terminals  M 
and  N.  The  currents  /,,  /a  ,.../„  in  each  branch  may  be 
constructed  as  in  Problem  IV.,  where  there  were  only  two 


RESISTANCE  AND  SELF  INDUCTION.  237 

branches,  and  the  resultant  current,  /,  in  the  main  line 
found,  since  it  is  the  geometrical  resultant  of  the  n  branch 
currents.  Fig.  58  is  constructed  as  follows.  Draw  a  semi- 


FIG.  58.— PROBLEM  V.  AND  PROBLEM  VI. 

circle  upon  the  impressed  E.  M.  F.,  OA,  and  lay  off  the  n 
different  angles  Bl ,  02 ,  .  .  .  Bn  in  the  negative  or  lag  direc-- 
tion,  which  represent  the  lag  of  the  current  in  each  branch 
behind  the  impressed  E.  M.  F.,  E.  This  will  give  n  differ- 
ent right  triangles  0 B A,  OCA,  0 D A,  etc.,  which  repre- 
sent the  E.  M.  F.'s  in  each  branch,  the  sides  of  which 
represent  the  effective  E.  M.  F.,  the  E.  M.  F.  to  overcome 
self-induction,  and  the  impressed  E.  M.  F.  Now  the  cur- 
rents /,,/,,  /3 ,  etc.,  or  0  F.  0  G,  OH,  etc.,  are  found  by 
dividing  the  effective  E.  M.  F.'s  R,  715  72272,  723  73 ,  etc.,  or 
OB,  6~C,  0~D,  etc.,  by  the  resistances  72, ,  722,  723 ,  etc. 
The  resultant  current,  /,  or  O  L,  is  found  by  taking  the 
geometrical  resultant  of  all  the  branch  currents,  0  F,  0  G, 
0  H,  etc.  This  construction  is  shown  by  the  closed  poly- 
gon 0  F  J  K  L,  each  side  of  which  is  equal  to  a  branch 
current.  By  this  construction  it  is  evident  that,  since  the 
angles  0  F  J,  F  J  K,  etc.,  must  be  each  greater  than  a  right 
angle,  the  maximum  resultant  current,  7,  or  O  L,  is  greater 
than  any  of  the  branch  currents.  During  a  certain  portion 
of  each  period,  as  before  explained,  the  instantaneous  value 


238  -       CIRCUITS  CONTAINING 

of  the  resultant  current  is  less  than  the  instantaneous  value 
of  the  current  in  any  one  branch. 

EQUIVALENT  EESISTANCE  AND  SELF-INDUCTION  OF  PARALLEL 

CIRCUITS. 

In  this  case,  as  in  the  previous  one,  suppose  that  a 
single  equivalent  circuit  is  substituted  for  the  n  parallel 
branches  having  such  a  resistance,  R  ',  and  self-induction, 
Lf,  that  the  current  in  the  main  line  is  not  changed  either 
in  magnitude  or  phase.  The  values  of  this  equivalent 
resistance,  R  ',  and  equivalent  self-induction,  Z',  may 
easily  be  found  in  terms  of  the  resistances  and  self-induc- 
tions of  each  branch.  In  Fig.  58  the  triangle  0  MA  must 
represent  the  triangle  of  E.  M.  F.'s  for  the  single  equivalent 
circuit  substituted  for  the  system  of  parallel  branches,  if 
-the  resultant  current  0  L  is  to  be  the  same  ;  for,  the  effect- 
ive E.  M.  F.,  R'  I,  is  in  the  direction  of  the  current  0  L,  and 
is  therefore  equal  to  O  M,  since  the  E.  M.  F.  to  overcome 
the  self-induction,  L'GJ  I  or  JO,  is  perpendicular  to  the 
current. 

To  find  R'  and  Z',  as  well  as  the  tangent  of  the  angle  & 
which  the  main  current  makes  with  the  impressed  E.  M.  F., 
we  may  proceed  as  folloAvs. 

If  we  take  the  projections  of  the  currents  /,  ll  ,  /,  ,  etc., 
upon  the  line  0  A,  we  obtain  the  equation 


(340)  I  cos  0  =  1,  cos  Ol  +  72  cos  6>2  +  .  .  .  =        /cos  0. 

If  we  consider  the  projections  of  the  currents  upon  a 
line  perpendicular  to  0  A,  we  obtain 

(341)  /  sin  0  =  I,  sin  0,  +  /,  sin  0,  +  .  .  .  =  2  /sin  6- 
Since   all   the  triangles  0  B  A,  OCA,  etc.,  are   right 


RESISTANCE  AND  SELF  INDUCTION.  239 

triangles,  the  following  values  for  Z,  Ilt  Z2 ,  etc.,  and  for 
cos  6,  cos  019  etc.,  sin  6,  sin  019   etc.,  will  be  evident. 

(342)  /=  E 


(344) 


E 

> 


^     etc* 


(343)  oos  0  = 


cos  ff,  = 


-— rr^fy        etc- 

Z'o? 


Z, 


Substituting  these  values  in  (340),  we  have 
7  cos  0  ^' 

(345)  -^r—  =  -pTT-j  — 7T1--. 


240  CIRCUITS  CONTAINING 

Making  a  similar  substitution  in  (341),  we  have 

f*Aa\      •/sin  0  L '  °° 

(346)      — ^T—  =  p/3|    r/2 — a 

ffj  _K       -+-   1  j       GO 

L,GO  L*  GO  f  ^^  L  GO 

-a  -r  7? 2  _i_ 


(348)  /         and  ^r^  -  B  «. 
Dividing  (346)  by  (345),  we  have 

(349)  tan  0  =  ^.  --  l^ 
From  equations  (345)  and  (346),  we  have 


-  T?,2  +  A2  <*'  r  E:  +  L?  «?  ^-R*+  Z2 

For  brevity,  let 


, 


Comparing  these  with  the  values  of  cos  0  and  sin  6  in  (343) 
and  (344),  we  obtain 


(350)  A  =  ,    or 


(351)   and        B  co  =  or     L>*= 

' 


RESISTANCE  AND  SELF  INDUCTION. 
For  cos2  0  and  sin2  0  we  may  substitute  the  values 
1  1  A* 


cos2  B  = 


1  +  tan2  d  JFe 

A1 

1  1 

sin2  6  = 


_L  cot2 


Making  these  substitutions,  equations  (350)  and  (351)  be- 
come 

A 


(352)  E'  = 


(353)  L'  00  = 


A*  +  B* 
.6ft? 


These  expressions,  (352)  and  (353),  enable  us  to  calculate 
the  equivalent  resistance  and  self-induction  of  any  number 
of  parallel  circuits  when  we  know  the  resistance  and  self- 
induction  of  each.  The  angle  of  lag  of  the  main  current  is 
found  from  (349).  These  same  analytical  results  were 
otherwise  obtained  by  Lord  Rayleigh,  and  given  by  him  in 
a  paper  on  "  Forced  Harmonic  Oscillations  of  Various 
Periods "  in  the  Philosophical  Magazine,  May  1886.  The 
present  demonstration  was  first  given  by  the  authors  in  the 
Philosophical  Magazine  for  September  1892. 

Problem  VI.    Divided  Circuit.    Current  Given. 

Suppose  we  have  a  number  of  circuits,  each  containing 
resistance  and  self-induction,  connected  in  parallel  as  in 
Fig.  57,  and  we  know  the  value  of  the  current,  7,  in  the 
main  line.  It  is  required  to  find  the  current  in  each  of  the 
several  branches.  The  value  of  the  impressed  E.  M.  F.  is 
not  known,  and  so  the  construction  cannot  be  made  in  the 
same  manner  as  in  the  problem  just  discussed. 


242  CIRCUITS  CONTAINING 

FIEST  METHOD.  ENTIKELY  GRAPHICAL. 
We  can,  however,  assume  any  value  for  the  impressed 
E.  M.  F.,  E,  and  make  the  construction  accordingly,  as  in 
the  previous  problem.  We  would  thus  obtain  a  value  for 
the  main  current,  /,  different  from  the  one  given.  The 
diagram  will  be  correct  in  all  respects  except  the  scale,  and 
this  must  be  changed  in  the  ratio  of  the  given  value  of  / 
to  the  value  of  /  obtained  from  the  assumed  impressed 
E.  M.  F.  The'  true  value  of  the  impressed  E.  M.  F.  and 
the  current  in  each  branch  may  thus  be  obtained  and  the 
solution  is  complete. 

SECOND  METHOD.     SOLUTION  BY  USE  OF  EQUIVALENT 
EESISTANCE  AND  SELF  INDUCTION. 

Another  solution  for  this  same  problem  is  obtained  by 
the  use  of  equivalent  resistance  and  equivalent  self-induc- 
tion of  parallel  circuits.  These  values  for  R'  and  L'  are 
calculated  according  to  the  expressions  (352)  and  (353). 
Draw  0  M,  Fig.  58,  equal  to  R'l,  and  draw  M  A  perpen- 
dicular to  O  M  and  equal  to  L'col.  The  hypotenuse  0  A 
of  the  right  triangle  O  MA  gives  us  the  value  of  the  im- 
pressed E.  M.  F.,  E.  The  further  construction  is  the  same 
as  before.  The  angles  of  lag  0, ,  09,  03,etc.,  are  laid  off, 
and  the  E.  M.  F.  triangle  for  each  branch  circuit  is  drawn. 
The  effective  E.  M.  F.  and  the  current  in  each  branch  are 
thus  readily  found. 

Problem  VII.    Effects  of  the  Variation  of  the  Constants 
M  and  L  in  a  Divided  Circuit  of  Two  Branches. 

RESISTANCE  ALONE  VARIED  IN  EITHER  BRANCH. 

Suppose  the  resistance  of  one  branch  of  a  divided  cir- 
cuit to  be  varied  and  the  other  constants  to  remain  un- 
changed ;  it  is  required  to  find  the  changes  in  the  currents 
due  to  this  variation  in  resistance  when  there  is  a  constant 


RESISTANCE  AND  SELF  INDUCTION. 


243 


impressed  E.  M.  F.  Let  the  diagram  for  the  divided  circuit 
shown  in  Fig.  55  be  represented  in  Fig.  59,  where  the  same 
letters  represent  the  same  points  as  in  the  diagram,  Fig.  56, 
already  given  for  the  divided  circuit. 

If  the  resistance  7?,  is  varied,  it  is  evident  that  the  effec- 
tive E.  M.F.  0~B  always  lies  on  the  semi-circle  0 B  A,  and, 
as  this  branch  may  be  regarded  as  a  single  circuit  having  a 
constant  E.  M.  F.  and  a  resistance  which  is  varied,  the  cur- 


FIG.  59.— VARIATION  OF  RESISTANCE  AND  SELF-INDUCTION  IN  A 
DIVIDED  CIRCUIT.    PROBLEM  VII. 

rent  fl  always  lies  on  the  semi-circle  ODIf,  whose  diameter 

E 

is  0  If  or  -j —  (see  PKOBLEM  I.).     If  R^  is  the  only  quantity 

varied,  it  is  evident  that  the  resultant  main  current  must 
lie  on  the  semi-circle  E  F  J,  whose  diameter  EJ  is  equal 
to  OH. 

Similarly,  if  R^  is  varied  alone,  the  current  7,  must  lie 


244  CIRCUITS  CONTAINING 

J7 

on  the  semi-circle  0 E K,  whose  diameter  is  OK  or  -f— . 

L^oo 

The  resultant  main  current  will  then  lie  on  the  semi-circle 
D  F L.  When  both  resistances  are  varied  at  the  same 
time  the  currents  ll  and  72  lie  on  their  semi-circles  0  D  H 
and  0 EK\  but  the  resultant  or  main  current  has  no  par- 
ticular locus. 

The  arrows  on  the  curves,  showing  the  effects  of  a  varia- 
tion of  the  resistance,  indicate  the  direction  of  the  change 
as  the  resistance  increases. 

SELF  INDUCTION  ALONE  VARIED  IN  EITHER  BRANCH. 

Regarding  each  branch  of  the  divided  circuit,  having  a 
constant  difference  of  potential  at  its  terminals,  as  a  single 
circuit,  it  is  evident  that  any  variation  of  Ll  alone  will  cause 
the  current  vector  II  to  lie  upon  the  semi-circle  0  D  M, 

Tfi 

whose  diameter  is  -=-  (see  PROBLEM  I.).     Any  variation  of 

Ll  alone  will  cause  the  resultant  main  current  vector,  /,  to 
lie  upon  the  semi-circle  E  F N. 

Similarly,  when  L^  alone  is  varied,  the  current'  /,  lies 
upon  the  semi-circle  OEP,  and  the  resultant  current  / 
upon  the  semi-circle  D  F  Q.  If  both  Ll  and  Lz  are  simul- 
taneously changed,  the  currents  /,  and  72  still  lie  on  their 
circles  0  D  M  and  0  E  P,  respectively,  but  the  resultant 
current  /  has  no  particular  locus. 

The  arrows  on  the  curves,  showing  the  effects  of  a  varia- 
tion of  the  self-induction,  indicate  the  direction  of  the 
change  as  the  self-induction  increases. 

LIMITING  CASES. 

This  diagram  enables  us  to  see  what  the  currents  will 
be  in  the  divided  circuit  in  the  limiting  cases  when  the 
resistances  or  self-inductions  approach  infinite  or  zero 
values.  As  a  particular  instance,  suppose  it  happens  that 


f 

RESISTANCE  AND  SELF  INDUCTION.  245 

Z2  is  zero,  and  Rl  is  very  small  compared  with  Lv.  This 
means  that  there  is  self-induction  alone  in  one  branch  and 
resistance  alone  in  the  other.  The  current  Jl  would  then 
be  represented  by  O  H,  and  /a  by  0  P,  and  the  main  cur- 
rent, /,  by  the  resultant  of  these. 

Constant  Potential  Example.— As  an  example,  suppose 
there  is  an  incandescent  lamp,  Fig.  60,  of  50  ohms  re- 
sistance 7?2,  and  a  coil  whose  self-induction  Z,l  is  .5 
henrys  shunted  around  the  lamp,  the  terminals  of  which 
are  subjected  to  a  constant  difference  of  potential  of  50 
volts.  What  are  the  currents  through  the  lamp,  coil,  and 


L, 
FIG.  60.—  PROBLEM  VII. 


the  main  line  ?    Suppose  that  a?  =  1000.     We  may  calcu- 
late 


#,""50 


,       E  50 

and         = 


E 


Make   0  P,  Fig.  61,  equal  to  —  =  1,  and  ninety  degrees 

E 

behind  it  make  OR—  T  — 

j^/l  GO 

O  S  is  easily  calculated,  thus  : 


E 

behind  it  make  OR—  T  —  =  .1.     The  resultant  current 


OS  =      ~OP*  +  O*  =  /I  +  .01  =  1.005,  approx. 


CIRCUITS  CONTAINING 


If  the  incandescent  lamp  should  break,  the  current  /7 
through  it  would  be  stopped  and  the  main  current  reduced 
to  OH,  equal  to  .1  amperes. 


FIG.  61.— CONSTANT  POTENTIAL  EXAMPLE,  PROBLEM  VII. 

Constant  Current  Example. — Suppose  that,  instead  of 
being  subjected  to  a  constant  potential,  a  divided  circuit,  as 
Fig.  60,  is  supplied  with  a  constant  current.  Let  the  main 
current  be  maintained  constantly  at  ten  amperes.  It  is  re- 
quired to  find  the  branch  currents  and  the  difference  of 
potential  at  the  terminals.  Let  it  have  a  resistance  7?2  of 
two  ohms,  and  let  the  self-induction  of  the  choke-coil  be.  02 


1O  Amperes 


FIG.  62.— CONSTANT  CURRENT  EXAMPLE,  PROBLEM  VII. 

hemys.  Using  the  first  method  of  solving  the  problem  of 
the  divided  circuit  when  the  current  is  given,  Problem  VI., 
we  may  assume  an  impressed  E.  M.  F.  0  A  of  ten  volts. 
Following  the  same  construction  in  Fig.  62  as  in  Fig.  61  for 


RESISTANCE  AND  SELF  INDUCTION.  247 

the  solution  of   tlie   constant  potential  example,  we  may 
calculate 


E  10 

and  '        =  =  "5  =  7'  = 


The  resultant  0  S  is  calculated  thus  : 

O~S  =  VlfP*  +7W  =  -y/25  +  .25  =  5.025  amperes. 

Since  the  main  current  should  be  ten  amperes,  it  is  neces- 
sary to   magnify  the  whole  diagram   in  the   ratio   g  AQg,  in 


order  to  find  the  true  difference  of  potential  at  the  ter- 
minals, and  the  true  branch  currents.     This   makes  the 

impressed  E.  M.  F.  10  X  K  noK  equal  to  19.9  volts  ;  the  cur- 


rent  7,  equal  to  9.95  amperes  ;  and  II  equal  to  .995  amperes. 
Since  the  current  and  the  E.  M.  F.  are  in  phase,  the  energy 
•consumed  by  the  lamp  is  equal  to  19.9  X  9.95  =  198  watts. 
TLe  energy  consumed  by  the  choke-coil  is*  almost  nothing, 
since  the  current  II  is  almost  at  right  angles  to  the  im- 
pressed E.  M.  F. 

If  the  lamp  filament  should  break,  the  current  72  would 
be  suddenly  stopped  and  the  whole  current  0  £  of  ten 
amperes  would  flow  through  the  coil.  The  potential  0  C  at 
the  terminals  would  suddenly  become  much  greater,  large 
enough  to  overcome  the  E.  M.  F.  of  self-induction  Ll  GO  I, 
that  is,  .02  X  1000  X  10  =  200  volts. 

Thus  the  choke-coil  shunted  around  the  lamp  consumes 
but  little  energy  and  prevents  the  current  from  being  inter- 
rupted when  the  lamp  breaks.  In  case  the  lamp  does 
break,  however,  there  is  the  sudden  rise  in  potential  as 
shown  above. 


CHAPTER  XVI.      v 

PROBLEMS  WITH  CIRCUITS  CONTAINING  RESISTANCE 
AND  SELF-INDUCTION. 

COMBINATION  CIRCUITS. 

Prob.  VIII.  Series    and    Parallel    Circuits.      Impressed  E.  M.  F.  given. 

Solution  by  Equivalent  JK  and  L. 
Prob.     IX.  Series  and  Parallel   Circuits.     Current  given.     Solution   by 

Equivalent  R  and  L. 

Prob.       X.  Extension  of  Problems  VIII  and  IX. 
Prob.     XI.  Series  and  Parallel  Circuits.     Entirely  Graphical  Solution. 
Prob.    XII.  Multiple  Arc  Arrangement. 

Problem  VIII.  Series  and  Parallel  Circuits.  Impressed 
E.  M.  F.  Given.  Solution  by  Use  of  Equivalent  Kesist- 
ance  and  Self  Induction. 

PROBLEMS  arising  from  combinations  of  series  and  paral- 
lel circuits  are  readily  solved  by  the  repeated  application 
of  the  foregoing  methods.  Let  us  consider  the  case  where 
two  systems  of  parallel  circuits  are  joined  in  series,  as  in 
Fig.  63.  The  resistance  and  self-induction  of  each  branch 
is  given  and  the  total  impressed  E.  M.  F.  It  is  required  to 
find  the  current  in  the  main  line  and  in  the  branches. 

The  equivalent  resistance  and  self-induction  Raf  and  Laf 
between  M  and  N,  and  Bb  and  Lbf  between  N  and  0,  are 
readily  found  according  to  the  formulae  (352)  and  (353).. 
We  can  now  treat  the  problem  as  that  of  a  series  circuit,  as 

348 


RESISTANCE  AND  SELF  INDUCTION. 


249 


in  PEOBLEM   III.,   and  ascertain  the   impressed  E.  M.  F. 
between  M  and  IT  and  between  N  and  0. 


FIG.  63.— PROBLEM  VIII.  AND  PROBLEM  IX. 


Upon  the  impressed  E.  M.  F.,  0  A,  Fig.  64,  draw  the 
right  triangle  OB  A  such  that  tan  A  OB  =  ^Z  ' 

•"'a. 


\ 


\ 


PIG.  64.— PROBLEM  VIII.  AND  PROBLEM  IX. 


Then  OB  is  the  E,  M.  F.  effective  in  overcoming  the  resist- 
ance Ea'  +  74'  and  may  be  divided  at  0  so  as  to  show  the 
E.  M.  F.  effective  in  overcoming  each. 


250  CIRCUITS  CONTAINING 


OCis  the  E.  M.  F.  effective  in  overcoming  the  resistance 
Ea  .     Draw  C  D  perpendicular  to  0  C\  and   complete  the 

right  triangle  0  CD,  so  that  tan  D  0  C  =  —  TTT-    O~D  is  the 

-Ka 

impressed  E.  M.  F.  between  the  points  M  and  N,  and  DA 
the  impressed  E.  M.  F.  between  N  and  0.  Knowing  the 
impressed  E.  M.  F.,  Ea  ,  between  M  and  N,  we  can  obtain 
the  currents  I,  and  72  in  the  branches  by  the  method  fully 
explained  in  PKOBLEM  IV.  and  PROBLEM  Y.  On  the  diameter 
O7),  the  E.  M.  F.  triangles  OFD  and  0  GD  are  drawn 
with  angles  of  lag  according  to  the  constants  of  each 
branch.  The  currents  /,  ,  7Q  ,  and  I  are  found  by  dividing 
the  effective  E.  M.  F.'s  by  the  resistances  Rl  ,  7?2  ,  and  Raf  , 
respectively.  In  the  same  way,  the  E.  M.  F.  triangles  D  LA 
and  D  P  A  are  erected  on  the  line  D  A,  which  represents 
Eb  ,  the  effective  E.  M.  F.  between  N  and  0.  The  currents 
J8  and  /4  are  then  found  and  we  have  the  complete  solution 
of  the  problem. 


Problem  IX.  Series  and  Parallel  Circuits.  Current 
Given.  Solution  l>y  Use  of  Equivalent  Resistance 
and  Self-induction. 

Suppose  that  we  have  the  same  arrangement  of  circuits 
as  that  just  described  and  shown  in  Fig.  63,  and  that  the 
main  current,  7,  is  given.  It  is  required  to  find  the  current 
in  each  branch.  The  solution  for  the  part  between  M  and 
N  and  for  the  part  between  N  and  0  can  be  obtained  inde- 
pendently according  to  the  second  method  given  in  PROB- 
LEM VI. 

In  Fig.  64,  O  C  is  drawn  equal  to  Ra'  '/,  and  CD  equal  to 
La'  GO  I.  On  0~D  the  E.  M.  F.  triangles  OFD,  OGD  are 
drawn  and  the  solution  for  branches  one  and  two  obtained. 


D  E  is  then  drawn  parallel  to  0  C  and  equal  to  Rb'  7,  and 
EA  equal  to  La'  GO  I.     The  E.  M.  F.  triangles,  D  L  A  and 


RESISTANCE  AND  SELF  INDUCTION. 


251 


D  P  A,  are  then  erected  on  D  A,  and  the  solution  for 
branches  three  and  four  obtained  in  the  regular  way.  The 
line  connecting  0  and  A  gives  the  total  impressed 
E.  M.  F.,  E.  . 

Problem  X.    Extension  of  Problems  VIII.  and  IX. 

The  solution  given  in  PROBLEM  VIII.  may  be  applied  to 
any  combination  of  circuits  in  series  and  parallel.  Let  us 
consider  a  combination  of  circuits  such  as  that  shown  in 


FIG.  65.-- PROBLEM  X. 

Fig.  65,  having  a  given  impressed  E  M  F  between  the  points 
JfaudP.  The  circuits  may  be  divided  into  three  parts, 
M N9  N  0,  and  0  P,  and  the  equivalent  resistance  and  self- 
induction  of  each  obtained  [see  (352)  and  (353)].  The  im- 
pressed E.  M.  F's.,  EaJ  Eb,  E09  of  each  portion  can  now  be 
laid  off,  Fig.  66,  according  to  the  method  given  for  series 


FIG.  66.— PROBLEM  X. 


circuits,  PROBLEM  III.,  and  a  semi-circle  erected  upon  each, 
as  was  done  upon  Ea  and  Eb  in  Fig.  64.    Each  portion  of  the 


252 


CIRCUITS  CONTAINING 


circuit  is  now  treated  separately  according  to  the  method 
for  parallel  circuits,  PEOBLEM  V.  In  each  semi-circle  the 
various  E.  M.  F.  triangles  are  drawn  and  the  currents  in  the 
several  branches  found. 

If  the  main  current  is  given  in  an  extended  system  of 
conductors,  as  in  Fig.  65,  the  solution  is  obtained,  as  in 
PEOBLEM  VII.,  by  dividing  the  system  into  its  several  sets 
of  parallel  circuits  and  treating  the  separate  sets,  M  N, 
NO,  OP,  independently. 

Problem   XI.     Series    and   Parallel    Circuits.     Entirely 
Graphical  Solution. 

In  the  previous  treatment  of  the  problems  arising  from 
combinations  of  circuits  in  series  and  parallel  it  was  neces- 
sary to  find  analytically  the  values  of  the  equivalent  resist- 
ance and  self-induction  of  each  set  of  parallel  circuits,  and 
the  solutions  were,  therefore,  partly  analytical  and  partly 
graphical.  They  may  be  obtained,  however,  by  entirely 
graphical  methods,  if  we  assume  some  value  for  the  current 
in  a  particular  branch  or  assume  its  impressed  E.  M.  F., 
solve  a  portion  of  the  system  of  conductors  accordingly,  and 
then  correct  the  scale  as  required  by  the  given  conditions  of 
the  problem.  Various  ways  of  doing  this  suggest  them- 
selves as  preferable  according  to  the  nature  of  the  problem. 

BY  ASSUMING  VAEIOUS  IMPEESSED  E.  M.  F.'s. 

Given  the  main  current,  /,  in  a  system  as  shown  in  Fig. 
67.  Let  us  assume  any  value  we  please  for  the  impressed 

E«    ..  ET,  Ec  E,  E, 


FlG.    67.--PROBLEM   XL 


E.  M.  F.,  Eb  ,  then  erect,  on  a  line  representing^,,  the  E. 
M.  F.  triangles  for  branches  one  and  two,  and   thus  find 


RESISTANCE  AND  SELF  INDUCTION. 


253 


the  currents  II ,  /3 ,  and  7,  due  to  this  assumed  E.  M.  R 
The  value  thus  obtained  for  the  main  current,  /,  will  be 
different  from  the  given  value,  and  the  assumed  E.  M.  F.,  Eb , 
must  be  changed  in  the  ratio  of  the  given  value  of  1  to  the 
value  obtained  from  the  assumed  value  of  Eb.  This  amounts 
to  changing  the  scale  of  the  drawing.  The  solution  is  thus 
obtained  for  the  part  between  I>  and  C.  The  several 
other  portions  C  D,  D  E  of  the  system  are  separately 
treated  in  the  same  manner  and  thus  a  complete  solution 
obtained. 

If  we  were  given  the  total  impressed  E.  M.  F ,  E,  and 
not  the  main  current,  /,  a  convenient  graphical  solution 
would  be  obtained  by  assuming  some  value  for  /,  solving 
as  in  the  previous  paragraph,  and  then  changing  the  scale 
according  to  the  ratio  of  the  given  value  of  the  impressed 
E.  M.  F.  to  the  value  thus  obtained. 

BY  ASSUMING  THE  CURRENT  IN  CERTAIN  BRANCHES. 

Instead  of  assuming  the  E.  M.  F.  impressed  at  the  ter- 
minals of  each  part  of  the  system,  we  may  assume  the  cur- 
rent flowing  in  any  one  branch  of  each  parallel  set  of  con- 
ductors. The  complete  graphical  solution  by  this  method 
of  a  combination  circuit  representing  in  Fig.  68  is  given, 

R,L, 


FIG.  68.— PROBLEM  XI. 


to  illustrate  the  principles  already  given.     The  total  im- 
pressed E.  M.  F.,  Et  is  given. 


254 


CIRCUITS  CONTAINING 


In  Fig.  69  any  assumed  line  0  A  is  drawn  to  represent 
the  current  in  the  branch  whose  resistance  and  self-induc- 


FIG.  69. —PROBLEM  XL 
THE  SOLUTION  FOR  CIRCUITS 
BETWEEN  A  AND  B,  FIG.  6 


FIG.  70.— PROBLEM  XI. 

THE  SOLUTION  FOR  CIRCUITS 

BETWEEN  C  AND  D,  FIG.  68. 


tion  are  Rl  and  Z, .  0  A  is  then  multiplied  by  Rl  and  pro- 
duced to  B.  Then  0~B  =  R,  I,  is  the  effective  E.  M.  F.  in 
branch  one.  Draw  B  C  perpendicular  to  0  B  in  the  direc- 
tion of  advance  and  make  it  equal  to  Z,  GO  Ir  Then  O  67is 
the  impressed  E.  M.  F.  necessary  to  drive  the  assumed 
current  through  branch  one.  Now  haying  this  impressed 
E.  M.  F.,  we  can  draw  the  E.  M.  F.  triangle,  O  D  A,  for 
branch  two,  and  obtain  the  current  0  .#  flowing  in  the  second 
branch  by  dividing  O  D  by  R^.  The  total  current  I  —  0  F 
is  then  the  vector  sum  of  7,  and  72 ,  or  of  0  A  and  O  E. 

For  the  parallel  system  between  1)  and  C  (Fig.  68),  the 
same  process  is  followed,  and  in  Fig.  70  O  A  is  first  assumed 
as  the  current  branch  in  three,  and  0  F  finally  found  to 
be  the  total  current  flowing  between  C  and  D  (Fig.  68). 
Since  these  two  parallel  systems  are  in  series,  the  total 
current,  OF  (Fig.  69),  flowing  between^  and  B  (Fig.  68), 
must  equal  the  total  current,  O  F(Fig.  70),  flowing  between 
C  and  D  (Fig.  68).  Hence  Fig.  70  is  magnified  until  1TF 
becomes  as  large  as  OF,  Fig.  69,  and  is  represented  in 
Fig.  71.  Next  the  two  figures  69  and  71  are  combined,  as- 


RESISTANCE  AND  SELF  INDUCTION. 


255 


shown  in  Fig.  72,  so  that  0'  F'  is  parallel  with  OF,  si$ce 
each  represents  the  same  current  /.     0  C ',  the  vector  sum  of 


FIG,  71.— PROBLEM  XI.    FIG.  70  ENLARGED. 

E—b  and  E^  ,  is  the  total  impressed  E.  M.  F.  at  the  termi- 
nals A  and  D  necessary  to  send  the  current  /.     If  this 


FIG.  72.— PROBLEM  XI. 


figure  is  now  magnified  until    0  C'  is  equal  to  the  given 
impressed  E.  M.  F.,  the  solution  of  the  problem  is  complete 


256  CIRCUITS  CONTAINING 

an$  we  have  found  the  currents  in  each  branch  for  the 
given  impressed  E.  M.  F. 

Problem  XII.    Multiple-arc  Arrangement. 

Graphical  solutions  for  circuits  in  series  and  for  circuits 
in  parallel  have  been  separately  explained  at  length  and  it 
has  been  shown  how  the  solution  of  any  combination  of 
circuits  in  series  and  parallel  may  be  obtained  by  dividing 
the  system  into  its  separate  parts  of  series  and  parallel  ar- 
rangements and  successively  applying  the  foregoing  meth- 
ods. There  are  countless  combinations  which  might  arise, 
but  the  solutions  of  all  depend  upon  the  principles  already 
given,  and  it  will  suffice  to  further  illustrate  them  by  their 
application  to  one  more  problem  of  combined  circuits. 

Let  us  consider  a  system  of  parallel  circuits,  each  with 
resistance  and  self-induction,  extending  between  two  mains 
containing  resistance  and  self-induction.  Such  a  system  is 
shown  in  Fig.  73.  The  circuits  1,  2.  3,  etc.,  contain  resist- 


FIG.  73.— PROBLEM  XII. 


ance  and  self-induction  72,  Z, ,  722  Z2 ,  Z?3Z3,  etc.,  respec- 
tively.    The  resistance  and  self-induction  of  the  mains  are 


/»  foR  fe 

>o    >o  >=> 

>C-3      >S    ,  >0 

a    ^rL  *? 


^^^- 


FIG.  74.—  PROBLEM  XII. 


Ra  and  La  for  the  portion  a,  Bb  and  Lb  for  the  portion  b, 
between  circuits  1  and  2,  Rc  and  Lc  for  the  portion  c,  etc. 


RESISTANCE  AND  SELF  INDUCTION.  257 

The  equivalent  resistance  and  self-induction  of  circuit  one 
and  that  portion  of  the  system  beyond  circuit  one — namely, 
Z>,  c,  d,  etc.,  and  2,  3,  4,  etc. — is  R  U  ;  for  circuit  two  and 
the  portion  of  the  system  beyond,  the  equivalent  resistance 


wrmim 
OQ     1xaQ3    f  lV_g   J    J?J 


FIG.  75.— PROBLEM  XII. 

and  self-induction  are  R"  and  L" ;  for  circuits  three  and 
beyond  they  are  R"  and  L'" ;  etc.  These  values  of  the 
equivalent  resistance  and  self-induction  are  computed  by 
successive  applications  of  the  formulae  (352)  and  (353),  be- 
ginning at  the  most  distant  end  of  the  system.  The  equiva- 
lent resistance  R""  and  self-induction  L""  are  found  by 
adding  7?6  and  Z5  to  Re  and  Le ,  respectively,  and  finding  the 
equivalent  resistance  and  self-induction  when  combined 
in  parallel  with  circuit  four.  R'"  and  U"  are  found  by 
adding  R"  '  and  L""  to  Rd  and  Ld  and  finding  the  equiv- 
alent resistance  and  self-induction  of  this  when  combined 
in  parallel  with  circuit  three.  R"  and  L",  and  R  and  L', 
are  similarly  found. 

Let  us  now  replace  by  a  simple  circuit  with  resistance 
and  self-induction  R'  and  L'  that  part  of  the  system  to 
which  it  is  equivalent.  The  system  then  reduces  to  a  series 
circuit  (Fig.  74),  and  its  solution  is  obtained  by  the  method 
for  series  circuits,  PROBLEM  III.  The  complete  solution  for 
this  problem  is  given  in  Fig.  76. 

Draw  ~0~A  —  K     On  0~A  erect  the  right  triangle  0  B  A 

so  that  tan  A  0  B  =  -j£ r>T  <»•     Find  the  point   G  such 

that 

::R'.Rr. 


258 


CIRCUITS  CONTAINING 


Draw  D  C  perpendicular  to  O  B,  and  complete  the  triangle 
0  CD  so  that  tan  D  0  C  =  -~  GO.    Then  Ea  represents  the 


FIG.  76.— PROBLEM  XII. 

impressed  E.  M.  F.  of  the  portion  a  of  the  circuit,  and  El  the 
impressed  E.  M.  F.  of  the  remaining  portion. 

L' 
tan  AD  E  =  JY,GO. 

Now  let  us  take  the  system  as  originally  shown  in  Fig. 
73,  and  replace  by  a  simple  circuit  with  resistance  R"  and 
self-induction  L"  that  part  of  the  system  to  which  it  is 
equivalent.  The  system  then  reduces  to  the  form  shown 
in  Fig.  75.  The  construction  of  Fig.  76  is  continued  as  be- 
fore. 

On  DA,  which  represents  Et ,  the  E.  M.  F.  impressed  at 
the  terminals  of  the  two  parallel  circuits,  draw  the  right 
triangle  D  F  A  so  that 

A  +  L" 


Divide  D  F  at  G  so  that 


DG  :  GF::  7?h  :  E". 


RESISTANCE  AND  SELF  INDUCTION.  259 

Construct  the  right  triangle  H D  G  so  that 

tan  H  D  G  =  ^GO. 
A& 

Then  Eb  is  the  E.  M.  F.  impressed  in  the  portion  b  of  the 
circuit,  and  Et  that  impressed  on  the  part  of  the  circuit 
beyond  b. 

Eepeated  applications  of  this  method  of  construction 
finally  give  the  complete  solution  of  the  problem,  and  we 
liave  E19  E^,  ^3,  etc.,  as  the  E.  M.  F.'s  impressed  on  the 
circuits  1,  2,  3,  etc. ;  and  Ea ,  Ebt  Ec,  etc.,  as  the  E.  M.  F.'s 
impressed  on  the  portions  a,  b,  c,  etc. 

Knowing  the  impressed  E.  M.  F.  on  any  simple  portion 
of  the  circuit,  a  triangle  of  E.  M.  F.'s  can  be  drawn  and  the 
current  obtained.  The  E.  M.  F.  triangles  on.Ea,  Eb,  and  Ec 
are  already  drawn  and  the  effective  E.  M.  F.'s,  Ra  la  ,  Rblb , 
RCIC)  found.  Thet  current  is  found  by  dividing  the- effec- 
tive E.  M.  F.  by  the  resistance.  In  a  similar  way  the  cur- 
rent in  each  of  the  branch  circuits  1,  2,  3,  etc.,  may  be 
found.  For  instance,  on  Ez  the  E.  M.  F.  triangle  0  L  N  is 
drawn.  The  effective  E.  M.  F,  LN,  divided  by  the  resist- 
ance gives  the  current,  /3. 

The  solution  of  this  problem  by  entirely  graphical 
methods  could  be  gone  through  with,  as  in  some  of  the 
previous  problems,  and  likewise  the  problem  of  the  same 
arrangement  of  circuits  with  the  current  in  some  portion  of 
the  circuit  given. 


CHAPTER  XVII. 

PROBLEMS  WITH  CIRCUITS  CONTAINING  RESISTANCE  AND 

SELF  INDUCTION.     MORE   THAN   ONE   SOURCE  OF 

ELECTROMOTIVE  FORCE, 

Prob.  XIII.  Electromotive  Forces  in  Series. 

Prob.  XIV.  Direction  of  Rotation  of  E  M.  F.  Vectors. 

Prob.    XV.  Electromotive  Forces  in  Parallel. 

Prob.  XVI.  Electromotive  Forces  Laving  Different  Periods. 

Problem  XIII.    Electromotive  Forces  in  Series. 

SUPPOSE  that  in  different  parts  of  a  single  circuit  there 
are  two  sources  of  harmonic  E.  M.  F.  It  is  required  to 
find  the  current  which  flows  and  the  various  falls  of  poten- 
tial in  the  different  parts  of  the  circuit. 


E  D 

FIG.  77.—  PROBLEM  XIII. 


Let  the  circuit  be  that  represented  in  Fig.  77,  where  El 
and  E  are  two  different  sources  of  harmonic  E.  M.  F  of  the 


same  period.     Draw  the   lines  0  A  and  O  7?,  Fig.  78,  to 
represent  the  E.  M.  F.'s  E^  and  E^  respectively. 

The  total  E.  M.  F.  acting  in  the  circuit  is  the  geometric 
sum  of  0  A  and  0  1>,  that  is,  the  diagonal  0  C  (see  page  213). 

260 


RESISTANCE  AND  SELF  INDUCTION. 


261 


Kegarding  O  C  as  the  impressed  E.  M.  F.  in  a  single  circuit, 
whose  resistance  is  2  R,  and  self-induction  2  L,  we  may 


FIG.  78.— PROBLEM  XIII. 


construct  the  triangle  of  E.  M.  F.'s  and  thus  find  the  current. 
Make  the  angle  COD  equal  to  tan'1  ^~s^.  Then  ~OD 

equals  12  B,  and  I)  C  equals  IGJ  2  L.  Dividing  0  D  b y 
2  J?,  we  obtain  the  current  /,  or  0  E.  To  obtain  the  various 
falls  of  potential  between  the  points  A  B>  B  C,  and  ED 
(Fig.  77),  divide  0  D  at  F  and  G  into  parts  proportional  to 
7?, ,  Et ,  and  7?3 ,  and  D  (7  at  H  and  /into  parts  proportional 
to  Zj ,  Z3 ,  and  Z3.  This  determines  the  points  J  and  A^and 
thus  gives  the  falls  of  potential  0  J,  J K,  and  KG  for  each 
part  of  the  circuit. 

Problem  XIV.  Direction  of  Rotation  of  E.  M.  F.  Vectors. 

When  two  harmonic  E.  M.  F.'s  are  connected  in  series, 
as  in  the  preceding  problem,  the  question  may  arise  whether 
it  may  not  happen  that  the  vectors  representing  the  two 
E.  M.  F.'s  revolve  in  opposite  directions.  It  is  evident  that, 
if  they  should  revolve  in  opposite  directions,  the  resultant 


262 


CIRCUITS  CONTAINING 


at  any  instant,  instead  of  lying  on  a  circle,  lies  ^iipon  an 
ellipse  (Fig.  79).   Here  0  B  is  an  E.  M.  F.  vector  revolving 


FIG.  79.— PROBLEM  XIV. 


counter-clockwise,  and  0  A  one  revolving  with  the  same 
angular  velocity  in  the  opposite  direction.  The  resultant 
0  C  must  always  lie  upon  the  ellipse.  The  major  axis  has 
a  fixed  direction  01)  which  bisects  the  angle  between  OA 
and  O  B.  The  magnitudes  of  the  semi-major  and  the  semi- 
minor  axes  are  equal,  respectively,  to  the  arithmetical  sum 
and  the  arithmetical  difference  of  the  vectors  0  A  and  0  B. 
If,  instead  of  drawing  0  A  in  the  direction  indicated, 
we  had  drawn  it  in  the  position  0  G  (making  the  angle 
G  0  H equal  to  A  OH),  and  caused  it  to  revolve  counter- 
clockwise in  the  same  direction  as  O  B,  the  projections, 
0  //,  of  0  A  or  O  G  would  be  the  same  at  every  moment. 
Consequently  the  vector  O  G  revolving  counter-clockwise 
represents  the  same  E.  M.  F.  at  every  moment  as  the  vector 
O  A  revolving  clockwise,  and  may  therefore  be  substituted 


RESISTANCE  AND  SELF  INDUCTION. 


263 


for  it.  But  the  resultant  of  0  G  and  O  B  gives  O  7,  whose 
locus  is  a  circle.  Thus  the  projection  of  0  1  is  the  same 
.as  the  projection  of  O  (J,  and  the  ellipse  may  therefore  be 
replaced  by  the  circle. 

It  is  never  necessary,  therefore,  to  consider  vectors  re- 
volving in  opposite  directions,  for  a  vector  revolving  in  one 
direction  can  always  be  replaced  by  a  vector  revolving  in 
the  opposite  direction. 


Problem  XV.    Electromotive  Forces  in  Parallel. 

Suppose  that  in  each  branch  of  a  divided  circuit,  such 
as  that  represented  in  Fig.  80,  there  is  a  source  of  har- 
monic E.  M.  F.,  and  that  all  these  E.  M.  F.'s  have  the  same 
period  ;  it  is  required  to  find  the  currents  in  the  branches. 

The  currents  may  be  found  by  making  use  of  the  gen- 
eral principle*  that,  if  the  currents  due  to  each  E.  M.  F. 
acting  separately  can  be  found,  the  current  which  flows 
when  all  the  E.  M.  F.'s  are  acting  together  is  the  geometri- 
cal sum  of  all  these  partial  currents. 


FIG.  80.— PROBLEM  XV. 


To  find  the  currents  due  to  all  the  E.  M.  F.'s  acting  to- 
gether we  may  then  proceed  by  regarding  each  branch,  1, 
2,  and  3,  in  turn,  as  the  main  line  in  which  there  is  the  im- 
pressed E.  M.  F.,  and  the  other  branches  as  a  divided  circuit. 


*  See  Mascart  and  Joubert's  Electricity  and  Magnetism,  Vol.  1,  Art.  202. 


264  CIRCUITS  CONTAINING 

Then,  considering  E^  to  be  the  only  E.  M.  F.  acting,  the 
problem  of  finding  the  partial  currents  //,  /2',  and  /,'  is 
readily  solved  by  the  methods  already  given.  Next,  con- 
sidering EI  as  acting  alone,  we  may  find  the  partial  currents 
//',  /,",  and  /,",  and  finally  we  find  //",  /,'",  and  I,'",  due 
to  Ez  acting  alone. 

The  actual  currents  in  the  branches  7j  ,  /2  ,  and  J3  when 
all  the  E.  M.  F.'s  act  together,  by  the  principle  just  stated, 
must  be  equal  to  the  geometrical  sum  of  the  partial  cur- 
rents ;  that  is, 

/!  —  geometrical  sum  of  //,  //,  and  7/  ; 

T  —  "  "      "     7  "    T"   arid   T"  - 

±n  —  J1   ,  _/2  ,  ctuu  ^3     , 


Problem  XVI.    Electromotive   Forces  Having  Different 

Periods. 

Let  there  be  two  impressed  harmonic  E.  M.  F.'s  in 
series  having  periods  which  bear  a  ratio  of  three  to  one. 
It  is  required  to  find  the  resultant  impressed  E.  M.  F.  and 
the  current  that  flows  in  the  circuit, 


In  Fig.  81  let  0  A  represent  maximum  value  of  El ,  and 
0  B  that  of  E^ ,  they  being  in  the  ratio  of  one  to  two.  As 
O  A  revolves  around  its  circle  three  times  as  fast  as  0  B, 
0  A  arrives  at  O  C  when  0  B  arrives  at  0  D,  and  the  re- 
sultant 0  /^traverses  the  curve  E  F  G  H.  If  the  projection 
of  the  resultant  vector  0  F  is  taken  upon  the  axis  0  Y  at 
equal  intervals  of  time,  we  may  thus  plot  the  curve  of  re- 
sultant E.  M.  F.,  Fig.  82.  This  E.  M.  F.  curve  is  the  plot 
of  the  equation 

e  =  E^  sin  3  GO  t  -f-  E^  sin  GO  t. 

The  curve  is  composed  of  two  simple  harmonic  components. 

To  find  the  current  which  this  resultant  E.  M.  F.  causes 

to  flow,  it  is  only  necessary  to  find  the  currents  which  each 


RESISTANCE  AND  SELF  INDUCTION. 


265 


component  E.  M.  F.  acting  separately  would  cause,  and  then 
add  these  together  geometrically.  If  there  is  self-induction 
in  the  circuit,  the  tangent  of  the  angle  of  lag  of  the  com- 


Resuifc, 


FIG.  81.— PROBLEM  XVI. 


ponent  currents  behind  their  respective  E.  M.  F.'s  is 


OP£  be  the  E.  M.  F.  triangle  upon  E* ,  and  OJ  the 
current  72 .     J  must  lie  upon  the  semi-circle  0  J M,  whose 

W 

diameter  is  -=^-  (see  PROBLEM  I.).    The  angle  of  lag,  A  0  $, 

Li  GL?2 

of  the  component  current  due  to  the  E.  M.  F.,  El ,  is  now 
determined,  since  its  tangent  is  three  times  the  tangent  of 


B  0  P,  thus, 


GO, 


3  L 


Also  the  current  O  IT,  or  /, , 


R  R 

due  to  the  E.  M.  F.  0  A,  or  El ,  is  now  determined,  since 
-/Tmust  lie  upon  a  semi-circle  0 K N  whose  diameter  ON 

equals  t  of  OM.     For  O~N  =  ^L  =  -A— ;  and  OM  = 


266 


L 


CO. 


RESISTANCE  AND  SELF  INDUCTION. 


=        L,  and  thus  OM  =  6  'ON.   The  resultant  of  ~OK 


and  0  J  gives  (9  Z.  and  this  vector  always  follows  the  curve 
marked  "  Besultant  Current."  If  the  projection  of  0  L  upon 
the  axis  OY  is  taken  at  regular  intervals  as  L  moves 
around  its  curve,  we  may  obtain  the  current  curve  Fig.  82. 


FIG.  82.— PROBLEM  XVI. 

This  current  curve  is  composed  of  two  simple  harmonic 
curves  each  due  to  a  simple  harmonic  E.  M.  F.,  but  the  two 
component  current  curves  lag  behind  their  respective  com- 
ponent E.  M.  F.  curves  by  different  angles.  For  this  reason 
the  resultant  current  curve  is  not  symmetrical  with  the  re- 
sultant E.  M.  F.  curve. 


CHAPTER  XVIII. 

INTRODUCTORY  TO   CIRCUITS   CONTAINING  RESISTANCE 
AND  CAPACITY. 

CONTENTS: — Problems  with  H  and  G  analytically  and  graphically  analo- 
gous to  problems  with  R  and  L.  Triangle  of  E.  M.  F.'s  for  a  single 
circuit  Containing  resistance  and  capacity.  Impressed  E.  M.  F. 
Effective  E.  M.  F.  Condenser  E.  M.  F.  Direction  shown  from  differ- 
ential equations.  Graphical  representation.  Two  methods  used. 
First  method  (the  one  used  throughout  this  book),  employing  E.  M.  F. 
necessary  to  overcome  that  of  condenser.  Second  method,  employing 
E.  M.  F.  of  condenser  Further  identification  of  analytical  and 
graphical  relations.  Mechanical  analogue. 

WHEN  Chapter  III.,  giving  the  analytical  treatment  of 
circuits  containing  resistance  and  self-induction,  is  com- 
pared with  Chapter  V.,  which  gives  the  corresponding 
analytical  treatment  of  circuits  containing  resistance  and 
capacity,  the  similarity  leads  us  to  infer  that  the  graphical 
solutions  of  problems  will  be  very  analogous  in  the  two 
cases.  Although  the  analogy  is  very  close,  which  fact 
makes  it  much  easier  to  follow  the  solutions  for  resistance 
and  capacity  and  is  a  great  help,  yet,  in  many  respects,  the 
contrast  is  so  marked  that  it  is  considered  advisable,  in 
discussing  problems  with  circuits  containing  resistance 
and  capacity,  to  give  the  solutions  for  the  same  arrange- 
ment of  circuits  as  those  which  have  been  given  for  circuits 
containing  resistance  and  self-induction  in  the  previous 
pages,  in  order  that  the  points  of  similarity  and  difference 
may  be  clearly  understood. 

267 


268  CIRCUITS  CONTAINING 

TRIANGLE  OF  ELECTROMOTIVE  FORCES  FOR  A  SINGLE  CIRCUIT 
CONTAINING  EESISTANCE  AND  CAPACITY. 

In  Chapter  V.,  in  which  circuits  containing  resistance 
and  capacity  were  analytically  treated,  it  was  shown  [equa- 
tion (78)]  that  when  the  impressed  E.  M.  F.  is  harmonic, 
that  is, 

e  =  E  sin  GO  t, 

the  resulting  current  which  flows  is  also  harmonic  and  is 

. E_ 

(78) 


The  charge  of  the  condenser  is  likewise  harmonic  and 
is  [equation  (79)] 

E 


These  equations  for  the  current  and  charge  were  de- 
rived from  the  differential  equation  of  electromotive  forces 
which  may  be  written  in  any  of  the  forms  [see  (55)] 


fi 

' 


d* 


de          di 

=  ^~ 


Here   e  is   the   instantaneous  value   of  the  impressed 
E.  M.  F.  of  the  source  ;  R  i  is  that  part  necessary  to  over- 

fiat     q 

come  the  ohmic  resistance;  and  -—  ^  —  or  -^  the  E.  M.  F. 
necessary  to  overcome  the  E.  M.  F.  of  the  condenser. 


RESISTANCE  AND   CAPACITY. 


269 


Let  the  vector,  0  A,  Fig.   83,  represent  the   harmonic 
impressed  E.  M.  F.  of  the  source.     Then,  by  equation  (78), 


FIG.  83. — TRIANGLE  OP  ELECTROMOTIVE  FORCES. 
FIRST  METHOD — THE  ONE  USED   THROUGHOUT  THIS  BOOK — EMPLOYING 

E.  M.  F.  TO  OVERCOME  THAT  OF  THE  CONDENSER. 

we  see  that  the  current  must  be  represented  by  a  vector, 


OB,   in  advance  of  OA  by  an  angle    0,   or   tan"1  /y  p      . 

C  M  GO 

The  effective  E.  M.  F.,  being  equal  to  RI,  has  the  same  di- 
rection as  the  current  and  must  be  represented  by  a  vector 
0  C  equal  to  the  current  vector,  0  B,  multiplied  by  R. 
The  E.  M.  F.  to  overcome  that  of  the  condenser,  having  the 

instantaneous  value  -4; ,  is  at  right  angles  to  the  current, 

and  must  therefore  be  represented  by  the  vector  C  A  per- 
pendicular to  O  B. 

It  may  be  shown  that  this  E.  M.  F.    ~  is  at  right  angles 
io  the  current  by  the  preceding  equations,  thus : 

fidt 
V__J_ E 

C~       C 


270  CIRCUITS  CONTAINING 

To  simplify  this  expression  substitute 

E 


1  = 


and  6  —  tan"1      p     . 

The  equation  then  becomes 

(354)  -£-  =  ~  sin  \  GO  t  -f  8  -  90°1. 

O         C  Gtf 

This  equation  shows  that  the  E.  M.  F.,  -~,  to  overcome 
that  of  the  condenser  is  ninety  degrees  behind  the  current, 
and  that  the  maximum  value  of  this  E.  M.  F.  is  ~^— 

C GO* 

The  vector  (Fig.  83),  whose  length  is  j^-  ,  ninety  degrees 

{;  GO 

behind  the  current,  O~B,  therefore  represents  the  E.  M.  F. 
to  overcome  that  of  the  condenser. 

The  E.  M.  F.  of  the  condenser  is  equal  and  opposite  to 
that  which  is  necessary  to  overcome  it,-  and  is  consequently 
ninety  degrees  in  advance  of  the  current  represented  by  the 
vector,  A  0',  Fig.  84. 

THE  METHOD  TO  BE  USED  IN  THE  GRAPHICAL  SOLUTIONS  OF 
PROBLEMS  FOR  CIRCUITS  CONTAINING  KESISTANCE  AND 
CAPACITY. 

In  the  graphical  treatment  of  problems  with  circuits 
containing  resistance  and  capacity,  just  as  was  the  case  with 
circuits  containing  resistance  and  self-induction,  there  are 
two  methods  of  drawing,  each  equally  correct,  which  will, 
if  followed  throughout,  give  identically  the  same  results. 
These  two  methods  arise  according  to  whether  the  E.  M.  F. 
necessary  to  overcome  the  E.  M.  F.  of  the  condenser  is  con- 


RESISTANCE  AND   CAPACITY. 


271 


sidered,  or  the  E.  M.  F.  of  the  condenser.   The  first  method 
is  illustrated  by  Fig.  83  ;  the  second  by  Fig.  84. 

In  order  that  uniformity  may  exist  throughout  all  the 
diagrams  which  represent  cases  where  both  self-induction 
and  capacity  are  considered  in  circuit,  since  the  method  of 


FIG.  84. — TRIANGLE  OF  ELECTROMOTIVE  FORCES. 
SECOND  METHOD,  EMPLOYING  E.  M.  F.  OF  CONDENSER. 

drawing  was  adopted  which  considered  the  E.  M.  F.  neces- 
sary to  overcome  the  self-induction,  here  we  are  obliged  to 
adopt  that  method  which  employs  the  E.  M.  F.  necessary 
to  overcome  the  E.  M.  F.  of  the  condenser,  as  represented  in 
Fig.  83. 

That  the  construction  of  the  figures  fulfils  the  condi- 
tions expressed  by  the  current  equation  (78)  may  be  shown 
again  by  a  further  comparison  of  the  relations.  Thus  in 
Fig.  83  or  84  it  is  evident  that 

I 
AC 


tan  A  0  C  = 


oc 


El       CRoD 


=  tan 


and  this  corresponds  to  the  angle  of  advance.  Again,  the 
impressed  E.  M.  F.,  0  A,  being  the  hypotenuse  of  the 
triangle  0  A  C,  is  equal  to  the  square  root  of  the  sum  of 
the  squares  of  the  two  sides,  and,  therefore, 


OA  = 


CIRCUITS  CONTAINING 


that  is,       E  = 
and  /  — 


E 


This  is  seen  to  correspond  to  the  maximum  value  of  the 
current  given  in  equation  (78). 

MECHANICAL  ANALOGUE. 

That  the  E.  M.  F.  of  the  condenser  is  at  right  angles  to 
the  current  may,  perhaps,  be  best  understood  by  the  phy- 
sical conception  of  the  part  played  by  the  condenser  in  a 
circuit.  A  good  mechanical  analogue  of  the  condenser  is 
an  air-chamber,  as  represented  in  Fig.  85,  in  which  the  air 
is  first  compressed  and  then  expanded. 
The  piston  P  moves  back  and  forth,  with 
an  harmonic  motion,  we  will  say,  first 
compressing  and  then  expanding  the 
air  in  the  chamber.  When  at  its  central 
position,  the  air  is  at  the  atmospheric 
pressure.  The  current  may  be  repre- 
sented by  the  motion  of  the  piston,  or 
of  the  water  in  the  tube  which  trans- 
mits the  pressure  to  the  air-chamber. 
The  charge  of  the  condenser  may  be 
represented  by  the  volume  of  water 
which  enters  or  leaves  the  air-chamber, 
the  charge  being  taken  as  zero  when 
the  piston  is  at  its  central  position,  that 
is,  when  the  air  is  at  the  atmospheric 
pressure.  Considering  the  moment 
FIG.  85.— MECHANICAL  when  tlie  piston  is  in  the  central  posi- 
ANALOGUE  OF  A  Cox-  upward,  the  charge  is 

DENSER. 

zero,  and  the  current  is  a  maximum,  as 
here  the  piston  moves  with  its  maximum  velocity.    The  cor- 


f 
RESISTANCE  AND   CAPACITY.  273 

responding  points  on  the  curves,  Fig.  84,  are  Hand  K\  that 
is,  the  positive  current  is  represented  by  the  upward  motion 
of  the  piston.  When  the  piston  arrives  at  Q,  the  upper  end 
of  the  stroke,  the  current  is  zero  and  is  represented  by  the 
point  N  on  the  curve.  The  charge  is  here  a  positive  max- 
imum, and  during  the  previous  quarter  of  the  stroke  the 
compressed  air  has  exerted  an  outward  pressure,  cor- 
responding to  the  E.  M.  F.  of  the  condenser,  opposed  to 
the  current.  This  pressure  reaches -a  negative  maximum, 
together  with  the  charge,  when  the  current  is  zero.  This 
corresponds  to  the  point  M  on  the  curve.  During  the  re- 
turn of  the  piston  to  the  central  position,  both  the  current 
and  the  pressure  are  in  the  same  negative  direction  until 
the  current  becomes  a  negative  maximum,  at  the  central 
position,  where  the  pressure  becomes  zero  and  then  changes 
sign.  This  example  shows  how  the  pressure  exerted  by 
the  air,  corresponding  to  the  E.  M.  F.  of  the  condenser,  is 
just  ninety  degrees  in  advance  of  the  current.  The  pres- 
sure which  must  be  exerted  upon  the  piston  to  overcome  the 
pressure  of  the  air  chamber,  corresponding  to  the  E.  M.  F. 
necessary  to  overcome  that  of  the  condenser,  is  evidently 
equal  and  opposite  to  the  pressure  of  the  air-chamber,  and 
lags,  therefore,  ninety  degrees  behind  the  current.  As  be- 
fore explained,  Fig.  83  represents  the  manner  of  drawing 
when  the  E.  M.  F.  necessary  to  overcome  that  of  the  con- 
denser is  considered,  and  Fig.  84  when  the  E.  M.  F.  of  the 
condenser  is  considered. 


CHAPTER  XIX. 

PROBLEMS  WITH  CIRCUITS  CONTAINING  RESISTANCE  ANI> 

CAPACITY. 

Prob.      XVII.  Effects  of  the  Variation  of  the  Constants  R  and  C  in  a 

Series  Circuit.    R  varied.     C  varied. 
Prob.    XVIII.  Series  Circuit.     Current  given.     Equivalent  R  and  G  in 

Series. 

Prob.       XIX.  Series  Circuit.     Impressed  E.  M.  F.  given. 
Prob.         XX.  Divided    Circuit.     Two    Branches.     Impressed    E.  M.  F. 

given.     Equivalent  R  and  G  for  Parallel  Circuit. 
Prob.       XXI.  Divided  Circuit.     Any  Number  of  Branches.     Impressed 

E.  M.  F.   given.     Equivalent   R  and    G  obtained   for 

Parallel  Circuits. 
Prob.     XXII.  Divided  Circuit.    Current  given.    First   Method:  Entirely 

Graphical.     Second  Method:  Solution  by  Equivalent  R 

and  C. 
Prob.    XXIII.  Effects  of  the  Variation  of  the  Constants  R  and   C  in  a 

Divided  Circuit  of  Two  Branches. 
Prob.    XXIV.  Series  and  Parallel  Circuits.     Impressed  E.  M.  F.  given. 

Solution  by  Equivalent  R  and  G. 
Prob.      XXV.  Series  and  Parallel  Circuits.     Current  given.     Solution  by 

Equivalent  R  and  G. 

Prob.    XXVI.  Series  and  Parallel  Circuits.    Entirely  Graphical  Solution. 
Prob.  XXVII.  Multiple-arc  Arrangement. 

Problem  XVII.    Effects  of  the  Variation  of  the  Constants 
It  and  C  in  a  Series  Circuit. 

THE  EESISTANCE  VARIED. 

WHEN  the  ohmic  resistance  is  varied  in  a  circuit  con- 
taining only  resistance  and  capacity,  the  current  is  changed 

274 


RESISTANCE  AND   CAPACITY. 


275 


and  it  is  of  interest  to  investigate  just  Low  it  changes 
both  in  magnitude  and  in  direction.  The  triangle  OAC, 
Pig.  86,  represents  the  triangle  of  E.  M.  F.'s  for  the  circuit 


CEw 

FIG.  86.  —  VARIATION  OF  RESISTANCE  AND  CAPACITY  IN  A  SERIES 
CIRCUIT.     PROBLEM  XVII. 

when  the  resistance  is  R.  The  current  O  B  is  equal  to  O  G 
divided  by  R.  Draw  a  line  0  D,  of  indefinite  length,  perpen- 
dicular to  the  E.  M.  F.  0  A  in  the  direction  of  advance. 
The  angle  D  0  C  is  the  complement  of  AOC,  and  is,  there- 
fore, tan"1  C  R  GO.  Draw  ^^perpendicular  to  OB  and  let 
it  meet  0  D  at  E.  The  line  B  Eileen  equals  G  R  GO  I;  for, 
OB  equals  /,  and  tan  E  0  B  equals  G  EGO. 

It  can  be  shown  that  the  hypotenuse  0  E  of  this  triangle 
is  equal  to  G  E  GO,  and  is  therefore  a  constant  entirely  inde- 
pendent of  any  variation  in  the  current  /,  or  resistance  R. 
Taking  the  square  root  of  the  sum  of  the  squares  of  the 
sides  0  B  and  B  E,  we  obtain 


Substituting  for  /  its  value 


E 


= ,  we  obtain 


I  Vl+C*  £*<*>*  =  CEGO, 
and,  therefore,  OW—  G  E  on. 


276  CIRCUITS  CONTAINING 


Now  since  the  side  OB  of  the  right  triangle  QBE  always 
represents  the  current  /,  and  the  hypotenuse  O  E  is  inde- 
pendent of  current  or  resistance,  it  follows  that  the  current 
is  always  represented  by  a  vector  O  B  inscribed  in  a  semi- 
circle QBE  for  any  possible  variation  in  the  resistance. 
In  the  figure  the  arrow  indicates  the  direction  of  variation 
as  the  resistance  increases. 

In  the  limiting  cases  when  R  is  infinite  or  zero,  we  see 
by  this  figure  the  limiting  values  of  the  current.  When  R 
is  infinite,  the  current  is  evidently  zero.  When  R  ap- 
proaches zero,  OB  approaches  0 E,  and  in  the  limit  the 
current  becomes 

7=  CEoo. 

When  the  circuit  contains  no  ohmic  resistance,  we  see,  first, 
that  the  impressed  E.  M.  F.  is  equal  to  -~— ,  the  E.  M.  F. 

O  GO 

of  the  condenser;  and,  second,  that  the  current  is  90°  in 
advance  of  the  impressed  E.  M.  F.  These  relations,  here 
geometrically  shown,  are  analytically  expressed  in  equa- 
tion (354). 

THE  CAPACITY  OF  THE  CONDENSES  VARIED. 

Suppose  that  the  capacity  of  the  condenser  in  the  cir- 
cuit is  varied  while  the  resistance  remains  the  same  ;  we 
wish  to  find  how  the  current  changes. 

In  the  same  figure,  86,  prolong  the  line  E  B  until  it  meets 
the  impressed  E.  M.  F.  O  A  prolonged  at  F.  Then  B  F 

equals  77-^  — ,  since  tan  B  O  F  equals  77-75 — . 
G  H  GO  C  R  GO 

T^he  hypotenuse  0  E  is,  therefore, 


C'' 


I 

RESISTANCE  AND  CAPACITY.  277 


From  the  value  for  1  in  (82)  it  follows  that 


Hence,  O  F  '  = 


Tjl 


Since  the  hypotenuse  O  F  is  independent  of  the  current 
/  or  the  capacity  C,  and  is  a  constant  for  any  variation  in 
C,  it  follows  that  the  current  is  always  represented  by  a 
vector  0  E  inscribed  in  the  semi-circle  0  B  F  for  any  pos- 
sible value  of  the  capacity.  In  the  figure  the  arrow  indi- 
cates the  direction  of  variation  as  the  capacity  increases. 

In  the  limiting  cases  when  C  is  zero  or  infinite,  we  see 
from  the  figure  the  value  of  the  current.  When  C  ap- 
proaches zero,  the  current  evidently  approaches  zero. 
When  G  approaches  infinity  (which  is  equivalent  to  having 
no  condenser  in  the  circuit),  the  current  vector  OB  ap- 


preaches  OF,  and,  in  the  limit,  1=  -Q,  and  the  current 

follows  Ohm's  law. 

That  the  construction  of  Fig.  86  is  consistent  with  the 
equations  is  further  shown  from  the  following  relations. 


(355)     WF=  (EE  +  WF)*  =     CR 


(356)     ~OE*+  ~OF*  =  C*  E*  «?+  —  =      (1  +  C*  I?  tf\ 


Equating  (355)  and  (356),  we  find 


T  T? 

-     1       C*  £•  af  =  E;    or    1= 


278 


CIRCUITS  CONTAINING 


a  result  which  is  identical  with  that  analytically  expressed 
in  equation  (82). 

Problem  XVIII.    Series  Circuit.    Current  Given. 

Let  there  be  a  circuit,  Fig.  87,  having  in  series  n  differ- 
ent resistances  Rlt  R^,  etc.,  and  n  condensers  with  ca- 
pacities (7j ,  6Y2 ,  etc.  It  is  required  to  find  the  impressed 
E.  M.  F.  necessary  to  cause  a  current  1  to  flow.  In  Fig. 
88,  make  0  A  equal  to  the  current  flowing.  Multiply  this 


FIGS.  87  AND  88  —PROBLEM  XVIII.  AND  PROBLEM  XIX. 

by  R, ,  and  lay  off  (TB  equal  to  E,  /,  which  is,  then,  the  ef- 
fective  E.  M.  F.  to  overcome  the  resistance  7^.  Draw  B  C 
perpendicular  to  ~0~A  in  the  negative  direction,  and  make 

the  angle  B  0  C  =  0,  =  tan  -1  c  R  ^    Then  B  0  C  is  the 

triangle  of  E.  M.  F.'s  for  that  part  of  the  circuit  between  A 
and  B,  Fig.  87. 

The  construction  of  the  figure  is  similar  to  that  of  Fig. 


RESISTANCE  AND   CAPACITY.  279 

54,  in  PROBLEM  II.,  but  differs  from  Fig.  54  in  the  fact  that 
the  E.  M.  F.  triangles  in  the  present  construction  are  so 
drawn  that  the  various  currents  are  in  advance  of  their  re- 
spective electromotive  forces.  The  triangles  C  I)  E,  etc., 
are  drawn  and  the  construction  completed  similar  to  the 
corresponding  case,  PROBLEM  II.,  of  a  series  circuit  with 
self-induction.  We  thus  find  the  impressed  E.  M.  F.  to 
be  OG. 

EQUIVALENT  BESISTANCE  AND  EQUIVALENT  CAPACITY  IN  SERIES. 

Suppose  that  we  replace  all  the  resistances  in  Fig.  87 
by  a  single  resistance,  and  all  the  condensers  by  a  single 
one  ;  it  is  required  to  find  that  resistance  and  capacity  which 
will  allow  the  same  current  to  flow. 

It  is  evident  that  if  ~0~K,  Fig.  88,  is  E'l,  and  K~G  is 

-777  —  ,  where  Rf  and  C'  represent,  respectively,  the  equiva- 


lent  resistance  and  equivalent  capacity,  the  same  current 


will  flow.     But   OK  =  /  2  R,   and  KG  = 

GO 

It  therefore  follows  that  R'  =  2  R,  and  ^  =  2  ^  • 

may  write  it  C'  =  —  ^   and  have  the  equivalent  capacity 

4 

•equal  to  the  reciprocal  of  the  sum  of  the  reciprocals  of  each 
.separate  capacity. 

Problem  XIX.    Series  Circuit.    Impressed  E.  M.  F. 
Given. 

The  circuit  being  the  same  as  in  Fig.  87  in  the  previous 
problem,  it  is  required  to  find  the  current  and  the  fall  of  po- 
tential through  each  of  the  various  parts  of  the  circuit  when 
the  impressed  E.  M.  F.  is  given.  From  the  remarks  on  equiv- 
alent resistance  and  capacity  immediately  preceding,  it  is 


280  CIRCUITS  CONTAINING 

evident  that  the  same  current  will  flow  if  these  equivalents 
are  substituted  for  the  separate  resistances  and  capacities. 
The  triangle  0  K  G  m  ij  now  be  drawn  and  the  current 
found.  From  this  point  we  may  proceed  as  in  the  preced- 
ing problem  to  find  the  various  falls  of  potential  0  G,  C  E, 
and 


Problem  XX.    Divided  Circuit,    Two  Branches. 
Impressed  E.  M.  F.  Given. 

Let  us  consider  the  problem  of  a  divided  circuit  having 
two  branches  in  parallel,  as  indicated  in  Fig.  89.     EacL 


FIG.  89.— PROBLEM  XX. 

branch  contains  resistance  and  capacity,  and  there  is  an 
impressed  E.  M.  F.,  E,  between  the  terminals  J/and  N ;  it 
is  required  to  find  the  main  current,  7,  and  the  currents  II 
and  /2  in  the  branches. 

This  problem  corresponds  very  closely  to  PROBLEM  IV., 
in  which  case  the  branches  contain  self-induction  instead 
of  capacity.  Fig.  90  represents  the  solution  of  the  present 
problem,  and  corresponds  to  Fig.  56,  PROBLEM  IV.  The 
difference  is  that  the  E.  M.  F.  triangles  0  B  A  and  0  C  A, 
Fig.  90,  lie  on  the  positive  or  advance  side  of  the  impressed 
E.  M.  F.  OA,  instead  of  on  the  negative  side  as  in  Fig.  56. 
Otherwise  the  construction  by  which  we  obtain  the  two 
currents  0  D  and  0  E,  and  the  resultant  main  current, 
0~F,  is  identical  with  that  in  PROBLEM  IV. 


RESISTANCE  AND  CAPACITY.  281 

EQUIVALENT  KESISTANCE  AND  CAPACITY. 

X 

Suppose  that,  instead  of  the  two  parallel  branches  just 
considered,  a  single  circuit  be  substituted  for  them  whose 
resistance,  R'y  and  capacity,  Cf,  is  such  that  the  same  cur- 


FIG.  90.— PROBLEM  XX. 

rent  as  before  will  flow  in  the  main  line.  The  triangle  of 
E.  M.  F.'s  for  this  equivalent  circuit  must  be  0  G  A,  Fig. 
90,  since  the  impressed  E.  M.  F.  is  0  A,  and  the  effective 
E.  M.  F.  is  O  G  in  the  direction  of  the  current,  and  the 
E.  M.  F.  G  A,  to  overcome  that  of  the  condenser,  is  at  right 
angles  to  the  current.  We  may  write,  therefore,  R'l  for 


0  G,  and  777-  for  G  A.      This    equivalent   resistance    and 

C    GO 

capacity  may  be  expressed  in  terms  of  the  resistances  and 
capacities  of  the  branches,  but  the  determination  of  these 
values  will  be  deferred  until  after  the  discussion  of  the  fol- 
lowing problem. 


282 


CIRCUITS  CONTAINING 


Problem  XXI.    Divided  Circuit.    Any  Number  of 
Branches.    Impressed  E.  M.  F.  Given. 

Let  the  divided  circuit,  M N,  Fig.  91,  have  n  branches 
in  parallel,  each  containing  resistance  and  capacity,  with  an 
impressed  E.  M.  F.,  E9  between  the  terminals.  It  is  re- 
quired to  find  the  main  current  /. 


FIG.  91.— PROBLEM  XXI.  AND  PROBLEM  XXII. 

The  construction  of  Fig.  92  is  similar  to  that  of  Fig.  58, 
in  PKOBLEM  V.,  except  that  the  E.  M.  F.  triangles  and  all 
the  branch  currents  are  laid  off  in  the  direction  of  advance 
.and  not  of  lag.  The  main  current  O  L  is  the  geometrical 
resultant  of  all  the  branch  currents  OF,  O  G,  OH,  and 
O  /,  as  before. 


FrG.  92.— PROBLEM  XXI.  AND  PROBLEM  XXII. 
This  diagram  gives  the  complete  solution  of  the  problem 
of  the  divide'd  circuit  containing  resistance  and  capacity. 
Here,  too,  as  was  the  case  with  the  divided  circuit  contain- 


RESISTANCE  AND  CAPACITY.  283 

ing  resistance  and  self-induction,  it  is  evident  that  the 
maximum  main  current,  7,  is  greater  than  any  of  the  branch 
circuits. 

EQUIVALENT  EESISTANCE  AND  CAPACITY  OF  PARALLEL  CIRCUITS. 

In  this  case,  as  in  the  previous  one,  we  may  suppose  a 
single  circuit  substituted  for  the  parallel  branches,  having 
such  a  resistance,  Rf,  and  capacity,  C ',  that  the  current  in  the 
main  line  is  not  altered  in  magnitude  or  phase.  The  values 
of  this  equivalent  resistance  and  capacity  in  terms  of  the 
resistances  and  capacities  of  the  branches  may  be  found  by 
proceeding  in  the  same  way  as  was  done  to  obtain  the 
values  of  the  equivalent  resistance  and  self-induction  of 
parallel  circuits,  PROBLEM  Y.  Equations  are  formed  by 
taking  the  projections  of  the  currents  first  upon  the  line 
O  A,  Fig.  92,  and  then  upon  a  line  perpendicular  to  0  A. 
In  these  equations,  values  for  /,  7, ,  72 ,  etc. ;  cos  0,  cos  0, , 
cos  02 ,  etc. ;  sin  0,  sin  0, ,  sin  02 ,  etc.,  obtained  from  the 
geometry  of  the  figure,  are  substituted,  and,  after  opera- 
tions similar  to  those  used  in  obtaining  equivalent  resist- 
ance and  self-induction,  the  following  expressions  are  ob- 
tained for  the  equivalent  resistance  and  capacity  of  parallel 
circuits. 

(356  a)  R'  =  A,    .^g,  ^ , 

(356  b)     and        -^  =  ^rt^p ' 

R 


where        A  = 


-, 

and 


284  CIRCUITS  CONTAINING 

The  main  current  is  in  advance  of  the  impressed  E.  M.  F. 
by  an  angle  0  such  that 

BCD 
tan  0  =  -  -r-. 


The  complete  proof  of  this  was  first  given  by  the  authors 
in  the  Philosophical  Magazine  for  September,  1892.  These 
results  may  be  obtained  from  the  general  expressions 
for  equivalent  resistance,  self-induction,  and  capacity  which 
are  discussed  in  PROBLEM  XXXI. 

Problem  XXII.    Dividetl  Circuit.    Current  Given. 

Suppose  that  we  have  a  number  of  circuits,  each  with 
resistance  and  capacity,  connected  in  parallel  as  in  Fig.  91, 
and  we  know  the  value  of  the  current  /  in  the  main  line. 
We  wish  to  find  the  current  in  the  several  branches.  There 
are  two  solutions  similar  to  the  two  given  for  the  corre- 
sponding case  of  circuits  with  self-induction. 

FIRST  METHOD.     ENTIRELY  GRAPHICAL. 

By  assuming  any  value  we  please  for  the  impressed 
E.  M.  F.,  E,  we  may  solve  as  in  the  foregoing  problem. 
The  scale  of  the  drawing  must  then  be  changed  in  the 
ratio  of  the  given  value  of  the  main  current,  /,  to  the  value 
of  1  thus  obtained  according  to  the  assumed  impressed 
E.  M.  F. 

SECOND  METHOD.     BY  USE  OF  EQUIVALENT  RESISTANCE  AND 

CAPACITY. 

The  problem  may  be  otherwise  solved  by  use  of  equiva- 
lent resistance  and  capacity  of  parallel  circuits  as  given  in 
(356  a)  and  (356  b).  OM,  Fig.  92,  is  laid  off  equal  to  R'l. . 


RESISTANCE  AND   CAPACITY. 


285 


The  line  M  A  is  drawn  perpendicular  to  O  M  and  equal  to 

7 
-777--      The   hypotenuse    0  A  is    the   impressed   E.  M.  F. 

The  further  construction  is  the  same  as  in  the  foregoing 
problem.  Upon  0  A  the  E.  M.  F.  triangle  for  each  branch 
is  drawn  and  the  current  and  angle  of  advance  found. 


Problem  XXIII.  Effects  of  the  Variation  of  the  Constants 
R  and  C  in  a  Divided  Circuit  of  Two  Branches. 

If  we  compare  PROBLEMS  I.  and  XVII.,  in  which  the  dis- 
cussion is  given  of  the  effects  of  the  variation  of  the  con- 
stants R  and  L,  and  /?  and  C  in  series  circuits,  we  see  that 


FIG.  98. — VARIATION  OF  RESISTANCE  AND  CAPACITY  IN  A  DIVIDED 
CIRCUIT.    PROBLEM  XXIII. 


the  two  problems  are  similar,  and  that  the  constructions  in 
Figs.  52  and  86  are  the  same  except   for  direction,  the 


286 


CIRCUITS  CONTAINING 


former  being  in  the  direction  of  lag  and  the  latter  in  the 
direction  of  advance.  The  present  problem  is  similar  to 
PROBLEM  YIL,  in  which  the  effect  of  the  variation  of  R  and 
L  in  a  divided  circuit  is  considered.  The  construction  is 
given  in  Fig.  93,  which  explains  itself,  and  is  exactly 
similar  to  that  given  in  Fig.  59,  which  was  fully  described 
in  PROBLEM  VII.  The  arrows  in  the  figure  indicate  the 
direction  of  the  charge  as  the  resistance  or  capacity  in- 
creases. 

Problem  XXIV.  Series  and  Parallel  Circuits.  Impressed 
E.  M.  F.  Given.  Solution  by  Use  of  Equivalent  Resist- 
ance and  Capacity. 

Problems  arising  from  combinations  of  series  and 
parallel  circuits  with  resistance  and  capacity  are  solved  by 
the  repeated  application  of  the  methods  used  for  the  fore- 
going simple  problems  in  the  same  way  as  were  solved  the 
problems  involving  combinations  of  circuits  with  resistance 
and  self-induction.  Let  us  consider  the  case  in  which  two 
systems  of  parallel  circuits  are  joined  in  series,  as  in  Fig. 
94.  The  resistance  and  capacity  of  each  branch  and  the 


FIG.  94.— PROBLEM  XXIV. 

total  impressed  E.  M.  F.  are  given.  It  is  required  to  find  the 
current  in  the  main  line  and  branches.  The  problem  is 
similar  to  PROBLEM  VIII.,  and  the  solution  given  in  Fig.  95 


t 

RESISTANCE  AND  CAPACITY. 


287 


is  obtained  by  a  construction  similar  to  Fig.  63.  The 
equivalent  resistance  and  capacity  Raf  and  Cd  between  M 
and  Nt  and  Rb  and  Cb  between  N  and  0,  are  calculated  ac- 
cording to  (356  a)  and  (356  b).  The  impressed  E.  M.  F.'s  Ea 
and  Eb  are  now  found  according  to  the  method  for  series 
circuits,  PROBLEM  XIX.  The  part  between  M  and  N  and 


FIG.  95.— PROBLEM  XXIV. 

the  part  between  N  and  0  are  now  separately  treated  by 
the  method  of  parallel  circuits,  PROBLEM  XXI.  The  con- 
struction is  shown  clearly  by  the  figure. 

A  more  extended  system  of  circuits  in  series  and  parallel 
is  solved  by  the  same  methods. 


Problem  XXV.  Series  and  Parallel  Circuits.  Current 
Given.  Solution  by  Use  of  Equivalent  Resistance 
ami  Capacity. 

Let  us  suppose  the  same  arrangement  of  circuits  as  that 
shown  in  Fig.  94,  and  that  the  main  current,  /,  is  given. 
It  is  required  to  find  the  current  in  each  branch.  The 
parts  between  M  and  N  and  between  N  and  0  may  be 
separately  solved  according  to  the  second  method  given  in 
PROBLEM  XXII.  The  solution  of  any  number  of  circuits  in 
series  and  parallel  could  be  readily  obtained  by  the  same 
method. 


288 


CIRCUITS  CONTAINING 


Problem  XXVI. 


Series  and  Parallel  Circuits.    Entirely 
Graphical  Solution. 


In  the  foregoing  treatment  of  problems  involving  series 
and  parallel  combinations  of  circuits  containing  resistance 
and  capacity  it  was  necessary  to  find  analytically  the 
values  of  the  equivalent  resistance  and  capacity  of  each  set 
of  parallel  circuits,  and  the  solutions  were,  therefore,  partly 
analytical  and  partly  graphical.  They  may  be  obtained,  as 
in  the  correspooding  cases  of  combinations  of  circuits  with 
resistance  and  self-induction  (see  PROBLEM  XI.),  by  entirely 
graphical  methods  by  assuming  the  value  of  the  current  in 
a  particular  branch  or  assuming  its  impressed  E.  M.  F. 
After  solving  in  this  way,  the  values  assumed  and  the  scale 
of  the  diagrams  must  be  altered  to  agree  with  the  given  con- 
ditions of  the  problem.  Figs.  96,  97,  98,  and  99  give  the 


FIGS.  96  AND  97.— PROBLEM  XXVI. 

construction  for  the  entirely  graphical  solution  of  two  par- 
allel sets  of  circuits  connected  in  series,  as  in  Fig.  94.  The 
method  is  to  solve  separately  each  parallel  set  of  circuits 
by  assuming  some  value  for  the  impressed  E.  M.  F.  or  for 
one  of  the  branch  currents.  Figs.  96  and  97  give  the  con- 
struction for  the  solutions  of  the  parts  M N  and  NO,  re- 
spectively, starting  with  assumed  values  for  the  branch 
currents  /t  and  73 .  Fig.  97  is  then  magnified,  as  shown  in 


RESISTANCE  AND   CAPACITY. 


289 


Fig.  98,  until  the  main  current  7  is  the  same  size  as  in  Fig. 
96.  The  two  figures,  96  and  98,  are  now  combined  in  Fig.  99 
so  that  O.Fis  parallel  to  0'  F',  since  each  represents  the 


FIG.  98  —PROBLEM  XXVI. 

current  /.  AVe  thus  find  E,  the  impressed  E.  M.  F.  which 
will  cause  the  current  /  to  flow.  If  the  value  of  the  im- 
pressed E.  M.  F.  had  been  given,  the  scale  of  the  diagram 


FIG,  99.— PROBLEM  XXVI. 

could  be  altered  until  E  equaled  the  given  value  of -the 
E.  M.  F.  The  figure  would  then  give  the  value  of  the  main 
and  branch  currents  which  flow  when  there  is  this  given 
E.  M.  F. 

Problem  XXVII.    Multiple-arc  Arrangement. 

Of  the  many  arrangements  in  which  circuits  with  resist- 
ance and  capacity  may  be  combined,  let  us  consider,  as  a 


290 


CIRCUITS  CONTAINING 


further  example,  the  arrangement  in  multiple  arc,  as  shown 
in   Fig.   100,     The   solution   is   obtained    by  dividing   the 


R,  c. 


E© 


Fm.  100. -PROBLEM  XXVII. 


system  into  different  parts  and  successively  applying  the 
foregoing  solutions  for  series  and  for  parallel  circuits.    This 


R«  c« 


FIG.  101.— PROBLEM  XXVII. 

problem  and  its  solution  are  similar  to  PROBLEM  XII.,  and 
it  will,  therefore,  suffice  to  merely  outline  the  method  to  be 


FIG.  102. -PROBLEM  XXVII. 

followed.  The  circuits  1,  2,  3,  etc.,  have  resistances  and 
capacities  R^ ,  R., ,  7?3 ,  etc.,  and  C19  6"Q ,  C^ ,  etc.  The  resist- 
ance and  capacities  of  the  mains  are  Rn  and  Ca  for  the  por- 
tion a ;  Rb  and  Cb  for  the  portion  b  between  circuits  1  and 
2;  7?c  and  Cc  for  the  portion  C',  etc.  R'  and  C'  are  the 
equivalent  resistance  and  capacity  for  circuit  1  and  the  part 
of  the  system  beyond,  as  indicated  in  Fig.  101.  R"  and 
C"  are  the  equivalent  resistance  and  capacity  for  circuit 


RESISTANCE  AND   CAPACITY. 


291 


2  and  the  part  of  the  system  beyond,  as  indicated  in  Fig. 
102.  Similarly,  E'",  R"",  G'",  C""  have  values  as  indi- 
cated. The  values  for  the  equivalent  resistances  and  ca- 
pacities are  found  by  the  successive  applications  of  the 
formulae  (356  a)  and  (356  b).  The  complete  solution  is  given 
in  Fig.  103,  and  its  construction  is  similar  throughout  to 


^  /J&M&B*-^?^ 


FIG.  103.— PROBLEM  XXVII. 

that  of  Fig.  76,  PROBLEM  XII.  E, ,  E, ,  E, ,  etc.,  give  the 
impressed  E.  M.  F.'s  of  the  several  parallel  branches.  By 
erecting  an  E.  M.  F.  triangle  on  each,  the  effective  E.  M.  F. 
and  so  the  current  in  each  branch  may  be  found  in  the 
usual  way.  Thus  in  branch  3,  Z  ^V  is  the  effective  E.  M.  F., 
and  73  the  current.  Ea ,  -&& ,  Ec ,  etc.,  give  the  impressed 
E.  M.  F.'s  in  the  portions  a,  b,  c,  etc.,  respectively,  and  the 
•currents  are  easily  found  from  the  effective  E.  M.  F.'s 
Ra  Ia ,  Rb  Ib  ,  Rc  lc  ,  etc.  The  full  construction  can  best 
be  followed  by  comparing  PEOBLEM  XII.,  the  similar  case 
of  circuits  with  resistance  and  self-induction. 


CHAPTER  XX. 

CIRCUITS  CONTAINING  RESISTANCE,  SELF-INDUCTION,  AND 

CAPACITY. 

CONTENTS  -.—Introductory.  Graphical  methods  for  circuits  with  R,  L,  and 
G  based  upon  graphical  methods  for  circuits  with  R  and  L,  and  R 
and  0.  Diagram  of  four  E.  M.  F.'s.  Triangle  of  E.  M.  F.'s.  Method 
consistent  with  analytical  results  obtained  for  circuits  with  R,  L,  and 
C.  Capacity  or  self-induction  which  is  equivalent  to  a  combination  of 
capacity  and  self-induction. 
Prob.  XXVIII.  Effects  of  the  Variation  of  the  Constants  in  Series  Circuit. 

R,  L,  C,  and  GO  varied. 
Prob.      XXIX.  Series  Circuit.     Current  given.     Equivalent  R,  L,  and  C 

of  Series  Circuit. 

Prob.        XXX.  Series  Circuit.     Impressed  E.  M.  F.  given. 
Prob.      XXXI.  Divided  Circuit.     Impressed  E.  M.  F.  given.     Equivalent 

R,  L,  and  G  of  Parallel  Circuits. 
Prob.    XXXII.  Example    of    a    Divided    Circuit.     Impressed   E.  M.  F. 

given. 

Prob.  XXXIII.  Divided  Circuit,     Current  given. 
Prob.  XXXIV.  Series  and  Parallel  Combinations  of  Circuits. 

IN  the  foregoing  chapters  the  complete  graphical  solu- 
tions have  been  given  for  any  combination  of  circuits  in 
series  and  parallel  when  the  circuits  contain  resistance  and 
self-induction  or  when  they  contain  resistance  and  capacity. 
In  the  first,  the  impressed  E.  M.  F.  of  the  source  is  equal 
to  the  E.  M.  E.  necessary  to  overcome  resistance  plus  the 
E.  M.  F.  necessary  to  overcome  the  counter  E.  M.  F.  of 
self-induction  ;  in  the  second,  the  impressed  E.  M.  F.  is 

292 


I 
RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY.     293 

equal  to  the  E.  M.  F.  necessary  to  overcome  the  resistance 
plus  the  E.  M.  F.  necessary  to  overcome  that  of  the  con- 
denser. In  each  of  these  cases  the  three  E.  M.  F.'s  were 
represented  by  the  three  sides  of  a  triangle. 

Where  a  circuit  contains  resistance,  self-induction,  and 
capacity  there  are  four  E.  M.  F.'s  to  be  considered.  The 
impressed  E.  M.  F.  is  equal  to  the  sum  of  the  E.  M.  F.'s 
necessary  to  overcome  the  resistance,  the  self-induction, 
and  the  condenser  E.  M.  F.,  respectively. 

The  E.  M.  F.  to  overcome  resistance  is  E  Z;  that  to 
overcome  the  self-induction  is  L  GO  I  and  is  90°  ahead  of 
the  current  ;  and  that  to  overcome  the  E.  M.  F.  of  the  con- 

denser is  -~  —  and  is  90°  behind  the  current.     These  may 


be  drawn  as  the  lines  0  A,  A  B>  and  B  (7,  respectively,  in 
Fig.  104,  and  the  geometrical  or  vector  sum  0  G  accord- 
ingly represents  the  impressed  E.  M.  F. 


FIG.  104.— DIAGRAM  OF  ELECTROMOTIVE  FORCES  IN  A  CIRCUIT  WITH 
RESISTANCE,  SELF  INDUCTION,  AND  CAPACITY. 

Now  the  E.  M.  F.  to  overcome  that  of  self-induction  and 
that  of  the  condenser  are  always  in  exactly  opposite  direc- 
tions, and  when  combined  give  one  E.  M.  F.  at  right  angles 
to  the  current.  Thus,  in  Fig.  104,  A  C  represents  the  com- 
bined effect  of  the  E.  M.  F.'s  L  GO  /and  T. — ,  represented  by 


294  CIRCUITS  CONTAINING 

A  B  and  B  (7,  respectively,  and  is  equal  to  ( —  L  00}  /. 

\C GO  I 

We  may,  therefore,  represent  the  E.  M.  F.'sin  a  circuit  con- 
taining resistance,  self  induction,  and  capacity  by  a  triangle 
whose  sides  represent,  respectively,  the  impressed  E.  M.  F., 
that  necessary  to  overcome  resistance,  and  that  necessary 
to  overcome  the  E.  M.  E.  of  self-induction  and  capacity 
combined.  Eig.  104  may  then  be  drawn  as  Fig.  105.  When 

TY-  is  greater  than  L  GO,  the  current  is  ahead  of  the  ini- 

C/  Ge7 

pressed  E.  M.  F.;  and  when  7^—  is  less  than  L  GO,  the  cur- 

U   GO 

rent  lags  behind. 


FIG.   105.—  TRIANGLE  OF  ELECTROMOTIVE  FORCES  IN  A  CIRCUIT  WITH 
RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY. 

The  tangent  of  this  angle  of  lag  or  advance  is 


oo 
-  - 


When  positive,  the  angle  is  one  of  advance  ;  when  negative, 
one  of  lag. 

The  impressed  E.  M.  F.  0~C,  being  equal  to  the  square 
root  of  the  sum  of  the  squares  of  the  two  sides  of  the  tri- 


RESISTANCE, SELF-INDUCTION,   AND   CAPACITY.      295 

But  this  radical  is  the  expression  called  the  impediment 
(see  page  131),  and  we  may  therefore  write 

E.  M.  F. 

Current  — 


Impediment* 

which  corresponds  to  Ohm's  law. 

We  have  now  established  the  graphical  method  of  repre- 
senting the  E.  M.  F.'s  in  a  simple  circuit  containing  resist- 
ance, self-induction,  and  capacity,  basing  it  upon  the 
graphical  solutions  already  given  for  circuits  containing 
resistance  and  self-induction,  and  circuits  containing  re- 
sistance and  capacity  alone.  These  were  separately  ob- 
tained from  the  analytical  equations  previously  given. 

Let  us  now  compare  these  graphical  methods  with  the 
analytical  results  obtained  in  the  discussion  of  circuits 
containing  resistance,  self-induction,  and  capacity.  The 
general  solution  for  current  in  a  circuit  with  an  harmonic 
impressed  E.  M.  F.  is  [see  (181)] 

E  .(...-./      1          Loo\  \ 


sin     GO t -l-tan' 


'This  shows  that  the  current  has  an  angle  of  lag  or  advance 
whose  tangent  is 


CRoj        E  ' 

the  angle  being  advance  when  positive  and  lag  when  nega- 
tive, which  corresponds  to  the  graphical  construction  just 
given.  The  maximum  value  of  the  current  is 

E  E 


- 

GO 


296  CIRCUITS  CONTAINING 

These  equations,  being  identical  with  those  just  obtained 
graphically,  show  that  the  analytical  results  are  correctly 
represented  by  this  graphical  method. 

CAPACITY  or*  SELF-INDUCTION  WHICH  is  EQUIVALENT  TO  A  COM- 
BINATION OF  SELF-INDUCTION  AND  CAPACITY. 

Let  C '  or  L'  denote  the  capacity  or  self-induction  which 
is  equivalent  to  a  given  combination  of  the  two,  that  is, 
which  allows  the  same  current  to  flow  in  the  circuit  when  it 
is  substituted  for  the  combination.  Keferring  to  Fig.  105,  we 

see  that  the  E.  M.  F.  of  the   combination  is  -^ L  GO  I. 

C  GO 

Regarding  this  as  a  positive  quantity,  i.e.,  supposing 
-~ —  >  L  GO,  we  may  put 

(j   GO 

II  11 

-  L  GO  /;  or  777-  =  Tr-  -  L  GO. 


C'GO  -  CGO  *  C'GO         Coo 

(357)        Hence     C' =  y-          ~  =  l  -  L  C  GO" 


which  is  positive  since  1  >  L  C  GO\ 

If  we  suppose  -^ —  <  L  GO,  then  L  GO  —  -~ —  is  positive. 

(j  GO  (_;   GO 

We  may  then  put  . 

L'  GO!  =  L  GO  I—  -^ — ;    or    U GO  =  L  GO  —  -Fr- . 

C  GO  C  GO 

(358)        Hence     L'  =  L  -  -^ — -z,  a  positive  quantity. 

L/    GO 


Problem  XXVIII.-Effects  of  the  Variation  of  the  Con- 
stants R,L,C9  and  GO.     Resistance  Varied. 

If  the  resistance  alone  be  varied  in  a  circuit  containing 
self-induction  and  capacity,  it  is  interesting  to  inquire  how 


RESISTANCE,  SELF-INDUCTION,   AND   CAPACITY.      297 

the  current  changes.  Since  the  combination  of  the  self- 
induction  and  capacity  is  equivalent  to  a  self-induction  or 
a  capacity,  we  may  substitute  this  equivalent  for  the  com- 
bination. The  change  in  the  current  caused  by  any  varia- 
tion in  the  resistance  must  therefore  be  the  same  as  that 
before  explained  (PROBLEMS  I.  and  XVII.)  in  the  case  of 
self-induction  or  capacity  alone. 


FIG.  106.— VARIATION  OF  CONSTANTS.     PROBLEM  XXVIII. 


In  Fig.  106,  0  (7  represents  the  impressed  E.  M.  F.,  Et 
divided  by  the  resistance  R.  The  current  I  may  either 
advance  ahead  of  or  lag  behind  0  C  according  to  whether 

-^ —  is  greater  or  less  than  L  GO.     For  certain  values  of 

O  GJ 

resistance,  self-induction,  and  capacity  let  0  A  represent 

the  current  in  advance  of  the  impressed  E.  M.  F.,  which 

1 
signifies  that  -~ —  >  L  GO.     Make  D  0  equal  to  C'  E  GO,  and 

C/  GO 

draw  the  semi-circle  0  A  D  upon  D  0  as  diameter.  This 
is  then  the  locus  of  the  current  vector  0  A  as  the  resistance 
alone  changes,  as  explained  in  Problem  XVII. 

Similarly,  if  the  quantity  77—  —  L  GO  had  been  negative 

(^  GO 

and  of  an  equal  magnitude  to  its  former  positive  value,  the 
current  would  have  been  represented  by  0  B  lagging  behind 
0  C,  and  any  variation  in  the  resistance  alone  would  cause 
the  current  vector  to  move  upon  the  semi-circle  QBE, 
being  equal  to  0  E  when  the  resistance  is  zero,  as  explained 


298  CIRCUITS  CONTAINING 

in  PBOBLEM  I.  It  is  to  be  noticed  that  Fig.  106  is  the  same 
as  Figs.  52  and  86  combined. 

The  arrows  It,  ./?,  show  the  direction  of  change  as  the 
resistance  increases. 

SELF-INDUCTION  OK  CAPACITY  VARIED. 
When  either  the  self-induction  or  capacity  alone  is  varied, 
it  is  evident  that  the  value  of  the  quantity  -^ L  GO  and, 

C  GO 

therefore,  the  value  of  the  equivalent  self-induction,  L ', 
or  equivalent  capacity  C',  is  changed.  Now  any  varia- 
tion in  the  equivalent  self-induction  will  cause  the  cur- 
rent vector  to  move  on  the  semi-circle  0  B  C,  as  explained 
in  PROBLEM  L,  and  any  variation  in  the  equivalent 
capacity  will  cause  the  current  vector  to  move  on  the 
semi-circle  0  A  (7,  as  explained  in  PROBLEM  XYII.  Any 
change,  then,  in  self-induction  or  capacity  will  cause  the 
current  to  move  through  some  part  of  the  circle  0  A  C  B, 

7j> 

whose  diameter  is  O  (7  equal  to  ^. 

The  arrow  Z,(7  shows  the  direction  of  change  as  the 
capacity  or  the  self-induction  increases. 

FREQUENCY  VARIED. 

When  the  frequency  of  alternation  is  varied,  it  is  equiv- 
alent to  a  variation  of  GO,  the  angular  velocity,  which  is 
equal  to  2  n  times  the  frequency.  Any  increase  in  the  fre- 
quency increases  the  effect  of  the  self-induction  or  the 
capacity.  If  the  self-induction  is  the  more  important  ele- 
ment and  the  circuit  has  an  equivalent  self-induction  [see 
equation  (358)], 

L'  =  L~     ~ 


RESISTANCE,  SELF-INDUCTION,   AND   CAPACITY. 

any  variation  in  the  frequency  will  cause  a  variation  in  the 
equivalent  self-induction  according  to  this  equation.  If 
the  capacity  is  the  more  important  element,  the  equivalent 
capacity  varies  with  GO  according  to  the  equation  (357), 

C'=         ° 


1-  LCoo*' 

It  has  just  been  shown  that  any  variation  in  the  equivalent 
self-induction  or  capacity  causes  the  current  vector  to  move, 
between  limits,  on  the  circle  0  A  C  B.  This,  then,  is  the 
effect  of  a  change  in  frequency.  The  direction  of  this 
change,  as  the  frequency  increases,  is  shown  by  the  arrow 
Z,  C  in  Fig.  106. 

Problem  XXIX.— Series  Circuit.    Current  Given. 

Let  there  be  a  circuit  having  n  different  coils  and  con- 
densers in  series  as  represented  in  Fig.  107.  It  is  required 
to  find  the  impressed  E.  M.  F.  necessary  to  cause  the  cur- 
rent /  to  flow,  and  the  difference  of  potential  at  the  termi- 
nals of  each  coil  and  condenser. 


R,L5 


'Q 

FIG.  107.— PROBLEM  XXIX.  AND  PROBLEM  XXX. 

In  Fig.  108  make  6^A  equal  to  the  given  current  /. 
Lay  off  6O?  equal  to  Rv  Z, ,  and  make  B  C  equal  to  Z,  GO  I 
perpendicular  to  0  B  in  the  positive  direction.  Then  in 

the   negative   direction   make  BD  equal  to    ^ The 

algebraic  sum  of  B~G  and  B~D  is  B~E.     Then  60T repre- 
sents the  potential  difference  between  M.  and  0,  Fig.  107 ; 


300 


CIRCUITS  CONTAINING 


O  (7,  the  potential  difference  between  M  and  N ;  and  B  I), 
or  C E,  the  potential  difference  between  N  and  0,  the 
terminals  of  the  condenser.  In  a  similar  manner  the  lines 
E  G,  El,  I K,  and  1 M  are  drawn  representing  the  poten- 


FIG.  108.— PROBLEM  XXIX  AND  PROBLEM  XXX. 

tial  difference  between  0  P,  0  Q,  QR,  and  Q  S,  respectively, 
Fig.  107,  until  we  finally  reach  the  point  M,  Fig.  108.  TTM 
is  then  the  required  impressed  E.  M.  F.  necessary  to  cause 
the  current  I  to  flow. 


EQUIVALENT  KESISTANCE,   SELF-INDUCTION,  OB   CAPACITY  OF 
SERIES  CIRCUITS. 

It  is  evident  from  Fig.  108  that  if  we  had  one  coil  only, 
whose  self-induction  L'  is  such  that  the  line  N  Mis  equal 
to  L '  GO  J,  and  whose  resistance  E'  is  such  that  the  line 
O  N  is  equal  to  R'  I,  the  same  current  /  would  flow  if  this 
coil  be  substituted  for  the  combination  of  condensers  and 
coils.  The  resistance  of  this  equivalent  coil  must  evi- 
dently be 

(359)  R  =  B,  +  R,  +  Rz  +  etc.  =  2  R. 


The  self-induction  of  the  coil,  being  represented  by  M  N 
divided  by  GO  /,  is  evidently  found  thus  : 


L'  &D!= 


RESISTANCE,  SELF-INDUCTION,  AND   CAPACITY.      301 
JOT  =  EH-  B^+J^G-  FTI+  TK-JL,   or 

/  I        •  I 


(360) 


If  it  happens  that  this  sum  is  a  negative  quantity,  the 
self-induction  cannot  replace  the  combination,  but  a  con- 
denser can.  It  will  be  seen  that  the  capacity  of  this  con- 
denser C'  may  be  found  as  follows  : 


(361)  Hence       C'  = 


GO    ^ 

J  GO 


These  equations,  (359),  (360),  and  (361),  give  the  means  for 
computing  the  equivalent  resistance,  self-induction,  and 
capacity  of  series  circuits. 

Problem  XXX.— Series  Circuit.     Impressed  E.  M.  F. 

Given. 

Suppose  the  impressed  E.  M.  F.,  represented  by  the 
line  0  J/,  Fig.  108,  is  given,  and  the  circuit  is  that  shown 
in  Fig.  107.  It  is  required  to  find  what  current  will  flow 
and  what  is  the  E.  M.  F.  at  the  terminals  of  each  coil  and 
condenser. 

If  the  equivalent  self-induction  L'  given  by  equation  (360) 
above,  or  the  equivalent  capacity  C'  given  by  equation  (361), 
is  calculated,  we  may  construct  the  triangle  of  E.  M.  F.'s 


302  CIRCUITS  CONTAINING 

0  M  N,  in  which    0  N  equals  12  R,  and  N  M  equals 

12  \L  GO  —  •7T—  ).     The  current  0  A  is  found  by  dividing 

\  O  GOl 

0  N  by  2  R.  After  we  have  obtained  the  value  of  the 
current,  we  may  proceed,  as  in  the  preceding  problem,  to 
find  the  E.  M.  F.  in  each  part  of  the  circuit. 

Problem   XXXI.— Divided   Circuit.     Impressed  E.  M.  F. 

Given. 

Let  us  consider  the  problem  of  a  divided  circuit  having 
resistance,  self-induction,  and  capacity  in  each  branch,  as 
shown  in  Fig.  109.  The  impressed  E.  M.  F.,  E,  is  given  ;, 


c, 


C9  L9          R3 

^ —   onrm-AAA 
FIG.  109.— PROBLEM  XXXI. 

it  is  required  to  find  the  main  and  branch  currents.  The 
construction  in  Fig.  110  gives  the  complete  solution.  Since 
the  impressed  E.  M.  F.  at  the  terminals  of  each  branch  is 
known,  each  may  be  separately  treated  as  a  simple  circuit 
containing  resistance,  self-induction,  and  capacity,  as  in 
PROBLEM  XXX.  Upon  0  A,  which  represents  the  impressed 
E.  M.  F.,  JE,  a  circle  is  drawn,  and  upon  0  A  the  several 
E.  M.  F.  triangles  0  B  A,  0  C  A,  0  DA,  are  erected  with 

angles    0l ,    02 ,    03 ,    of  advance  or  lag  according  as  -~ — 

C  GO" 

is  greater  or  less  than  L  GO  .  The  currents  Tlt  72 ,  73  are 
found  by  dividing  the  corresponding  effective  E.  M.  F.'s 
by  the  resistance  Rl ,  7?2 ,  Bs ,  respectively,  and  the  main 
current,  /,  is  found  by  taking  the  vector  sum  of  the  branch 
currents.  The  problem  is  in  every  way  the  same  as  the 


RESISTANCE,  SELF-INDUCTION,   AND   CAPACITY.      303 

problem  of  the  parallel  circuits  with  the  resistance  and 
self-induction,  or  with  resistance  and  capacity,  except  that 
the  current  in  any  one  branch  may  be  either  in  advance  or 
behind  the  impressed  E.  M.  F.,  according  to  the  particular 
values  of  the  resistance,  self-induction,  and  capacity  of  that 
branch. 

EQUIVALENT  EESISTANCE,  SELF-INDUCTION,  AND  CAPACITY  OF 
PAKALLEL  CIRCUITS. 

Let  us  suppose  that  for  the  parallel  system  there  is  sub- 
stituted a  simple  circuit  containing  resistance  and  self- 
induction,  or  resistance  and  capacity,  such  that  the  same 
main  current  will  flow.  The  investigation  of  the  values  of 
equivalent  resistance,  self-induction,  and  capacity  is  similar 


FIG.  110.— PROBLEM  XXXI. 


to  the   determination  of  equivalent  resistance   and   self- 
induction,  PROBLEM  V.,  and  first  appeared  in  a  paper  by 


304  CIRCUITS  CONTAINING 

the  authors  in  the  Philosophical  Magazine  for  September, 
1892. 

If  we  take  the  projections  of  the  currents  /,  II ,  /2 ,  etc., 
upon  the  line  0  A,  we  obtain  the  equation 

(362)  /cos  6=1,  cos  0,  +  Z,  cos  #2  +  .  .  .  =  2  /cos  0. 

If  we  consider  the  projections  of  the  currents  upon  a  line 
perpendicular  to  0  A,  we  obtain 

(363)  /sin  8  =  /,  sin  0,  +  72  sin  02  +  .  .  .  =  2  /sin  0. 

Since  all  the  triangles  DBA,   OCA,  etc.,  are  right  tri- 
angles, we  get  the  following  relations  : 

(364)  / 


^'  + 


(365)  cos  0  =  E 


cos  0,  -  --. -  etc. 


RESISTANCE,  SELF-INDUCTION,   AND   CAPACITY.      305 

1 
(366)  sin  0  = 


sin  0t  = 


Ln  GO 


sin  02  =  —          2  =,  etc. 


Substituting  these  values  in  (362),  we  have 

I  cos  6  R' 

(367)     —w-=- 


Making  a  similar  substitution  in  (363),  we  have 
<368)        E 


— LGO 

C  GO 


'^CT^w  =  Ew- 


306  CIRCUITS  CONTAINING 

Here  the  letters  A  and  B  are  introduced  to  simplify  the 
resulting  expressions. 

Dividing  (368)  by  (367),  we  have 

(369)  tan  0  =  ^ 

jA_ 

Comparing  (365)  and  (367),  we  obtain 

/Q«rm  A        G0^8  r>> 

(370)  A  =  —>~>     or     R  = 


Comparing  (366)  and  (368),  we  obtain 
(371)     J.-T^T.    or    ^  -*.  =  £ 

a'-^-L" 

For  cos'  0  and  sin2  0  we  may  substitute  the  values 

* 

~ 


tan2  0  ~         B'tf 
A* 


~ 


cot2  e  ~  n          A     ~  A*  +  B*  tf  * 

1  +  5r^ 

With  these  substitutions  equations  (370)  and  (371)  be- 
come 

A 


(372)  R'  =  -gf 


+ 


Here  J.  and  ^  G?  each  stand  for  a  summation,  as  ex- 
pressed in  (367)  and  (368),  and  are  calculated  from  the 
particular  values  of  the  resistance,  self-induction,  and 
capacity  of  each  branch.  This  gives  a  definite  value  to  the 
equivalent  resistance,  R',  according  to  (372),  and  a  de- 


RESISTANCE.  SELF-INDUCTION,   AND   CAPACITY.      307 


finite  value  to  -r~  ~  L'  G?,  according  to  (373).     There  may 


be  an  indefinite  number  of  values  assigned  to  L'  or  C' 
according  to  values  assigned  to  the  other,  that  is,  we  may 
assume  any  value  for  L'  and  by  (373)  determine  the  value 
for  67',  or  vice  versa. 

If  the  right-hand  member  of  (373)  is  positive,  we  may 
consider  that  the  equivalent  circuit  has  no  self-induction, 
i.e.,  L'  =  0,  and  calculate  the  equivalent  capacity.  If  this 
member  is  negative,  we  may  consider  that  the  equivalent 
•circuit  has  no  condenser,  i.e.,  C'  —  GO,  and  calculate  ac- 
cordingly the  equivalent  self-induction.  In  any  case, 
therefore,  we  may  speak  of  the  equivalent  resistance  and 
self-induction,  or  the  equivalent  resistance  and  capacity  of 
a.  combination  of  circuits,  according  as  the  equivalent 
simple  circuit  would  have  self-induction  or  capacity. 

The  angle  of  lag  or  advance  of  the  main  current  is  ob- 
tained from  equation  (369). 

BRANCH  CIRCUITS  WITH  KESISTANCE  AND  SELF-INDUCTION  ONLY. 

There  is  no  condenser  in  any  branch  and  the  capacity 
of  each  is,  therefore,  infinite.  We  can,  accordingly,  obtain 
the  expressions  for  A  and  B  GO  for  this  case  by  substituting 
C  —  oo  in  the  summations  in  (367)  and  (368).  This  gives 

E 


A  = 


L  GO 


-«^  IF  +  ZSaf' 

From  (372)  and  (373),  we  have 

fil"  ^'  =  ^W- 

—  BGO 


308  CIRCUITS  CONTAINING 

These  results  are  seen  to  be  identical  with  those  obtained 
in  PROBLEM  Y.  and  given  in  equations  (352)  and  (353). 


BRANCH  CIRCUITS  WITH  RESISTANCE  AND  CAPACITY  ONLY. 

In  this  case  there  is  no  self-induction  in  any  branch,  and 
the  expressions  for  A  and  B  GO  are  found  by  substituting 
L  =  0  in  the  summations  in  (367)  and  (368).  This  gives 


R 


1 

C  GO  ^-  C  GO 


The  expression  for  R'  is  the  same  as  that  in  (372),  and 
from  (373)  we  get  an  expression  for  the  equivalent  capacity, 
thus: 


C'  GO  ~  A*  +  B*  o*a ' 

These  results  are  identical  with  those  previously  given 
in  PROBLEM  XXI. 


Problem  XXXII.— Example  of  a  Divided  Circuit, 
Impressed  E.M.  F.  Given. 

Suppose  a  divided  circuit  has  a  condenser  with  a 
capacity  C  of  one  micro-farad  in  one  branch,  and  a  coil 
whose  self-induction  L  is  one  henry  and  resistance  R  one 


RESISTANCE,  SELF-INDUCTION,   AND   CAPACITY.      309 

hundred  ohms  in  the  other  branch,  as  in  Fig.  111.     Let 
the  impressed  E.  M.  F.  be  one  thousand  volts,  and  2  n  times 


C  =  1  Micro-Farad 


R=1OOOhms 


L=1  Henry 
1OOO  Volts 


FIG.  111.— PROBLEM  XXXII. 

the  frequency  be  one  thousand.     What  are  the  currents  in 
the  main  line  and  branches  ? 

Since  there  is  no  resistance  in  the  condenser  branch, 
the  current,  0  B,  Fig.  112,  is  ninety  degrees  in  advance  of 


CEW  =  1  Ampere 

FIG.  112.— PROBLEM  XXXII. 


the  impressed  E.  M.  F.,  O  A,  and  is  equal  to  C  E  GO  =  10  G 
X  1000  X  1000  =  1  ampere.     The  tangent  of  the  angle  of 

L  GO  1  X  1000 

lag  of  the  current  in  the  coil  is  —75-,  equal  to  — ^7^ —  =  10, 


IT1 


100 


and  therefore  the  current,  O  D,  in  the  coil  is  almost  ninety 
degrees  behind  the  impressed  E.  M.  F. 


310  CIRCUITS  CONTAINING 

The  current  0  D  is  almost  equal  to  one  ampere,  for 
is  almost  equal  to  0  G,  and 

E  1000 

=      ^^ 


"We  have,  then,  the  condenser  current  OB  and  the  coil 
current  0  D,  each  equal  approximately  to  one  ampere, 
one  in  advance  of  the  E.  M.  F.  and  the  other  lagging 
behind.  The  resultant  of  these  two  branch  currents  is 
OE  and  is  equal  to  one  tenth  of  an  ampere,  approx- 
imately ;  that  is,  each  branch  current  is  about  ten  times  as 
large  as  the  main  current.  In  this  case  the  main  current 
is  almost  in  phase  with  the  impressed  E.  M.  F.,  being  in 
advance  of  it  by  a  small  angle. 

Problem  XXXIII.  Divided  Circuit.    Current  Given. 

If  we  have  a  number  of  parallel  circuits,  containing 
resistance,  self-induction,  and  capacity,  and  know  the  value 
of  the  main  current,  7,  the  solution  is  similar  to  that  given 
in  PROBLEM  VI.  The  first  method  of  solution  consists  in 
assuming  an  impressed  E.  M.  F.,  solving  as  in  the  previous 
problem,  and  then  correcting  the  scale  to  agree  with  the 
given  value  of  the  current.  The  second  method  consists  in 
computing  the  equivalent  resistance  and  equivalent  self-in- 
duction or  capacity  of  the  parallel  system,  according  to  the 
formulae  (372)  and  (373),  finding  graphically  the  impressed 
E.  M.  F.,  and  then  solving  according  to  the  last  problem. 

Problem  XXXIV.—  Series  and  Parallel  Combinations  of 

Circuits. 

In  the  graphical  treatment  of  circuits  with  resistance 
and  self-induction,  and  of  circuits  with  resistance  and 
capacity,  the  discussion  was  given  first  of  series  circuits 


RESISTANCE,  SELF-INDUCTION,  AND  CAPACITY.      311 

and  then  of  circuits  connected  in  parallel.  It  was  then 
shown  how  problems  arising  from  any  combination  of 
circuits  in  series  and  parallel  could  be  readily  solved  by 
repeated  applications  of  the  methods  given  for  the  solution 
of  series  and  parallel  circuits.  In  the  problems  given  for 
circuits  containing  resistance,  self-induction,  and  capacity 
the  full  solution  has  been  given  for  series  and  for  parallel 
circuits.  These  principles  may  be  applied  in  solving  any 
combination  of  series  and  parallel  circuits,  and  to  go 
through  particular  examples  of  these  would  be  needless. 
The  same  problems  as  those  given  fora  circuit  with  re- 
sistance and  self-induction  or  capacity  can  be  solved  in  the 
same  way  if  the  circuits  contain  all  three.  The  problems 
given  have  been  selected  as  examples  and  not  as  exhaus- 
tively representing  all  the  problems  which  these  graphical 
methods  are  adapted  to  solve.  The  various  combinations 
which  arise  are  endless  and  may  often  be  solved  in  more 
ways  than  one.  The  choice  of  method  depends  upon  the 
particular  requirements  of  the  problem.  A  clear  idea  of 
the  principles  involved  in  the  simple  cases  will  enable  one 
to  extend  them  with  ease  to  whatever  problems  arise. 


APPENDIX  A. 

EELATION  BETWEEN  PKACTICAL  AND  C.  G.  S.  UNITS. 

ELECTRICAL   UNITS. 


Practical 

< 

3.  G.  S.  System. 

System. 

Electro- 
magnetic. 

Electrostatic. 

Quantity  

1  coulomb 

10-1 

v  X  10'1  —  3  X  10* 

Current  

1  ampere.. 

IO-1 

v  X  10'1  —  3  X  IO9 

Potential  

1  volt 

108 

IO8  —  v  —  *  X  10~2 

Resistance  

1  ohm  .   .  . 

109 

Capacity    

1  farad  .  . 

io-9 

v'  X  10-  9—  9  X  IO11 

Self-induction  

1  henry 

109 

Mutual  induction. 

1  henry.  .  . 

IO9 

(v  =  velocity  of  light  =  3  X  IO10.) 


MECHANICAL  UNITS. 

Practical  System.  C.  G.  S.  System. 

Unit  length  .................  =  IO9  cm. 

Unit  mass  ..................  =  IO"11  grms. 

Unit  time  ..................  =1  sec. 

One  joule  ...................  =  IO7  ergs. 


One  watt  = 


h.  p  ..........  =  IO7  ergs  per  sec. 

312 


APPENDIX  B. 

SOME  MECHANICAL  AND  ELECTBICAL  ANALOGIES. 

TABLE  I.  —  LINEAR  MOTION. 

Notation. 

1.  Time  =  t. 

2.  Distance  =  s. 

ds 

3.  Linear  velocity  =  v  =  -TJ;    or,     ds  =  vdt. 

dv      d*  s 

4.  Linear  acceleration  =  a  =  -77  =  -j-^. 

5.  Mass  =  M. 

6.  Momentum  =  Mv. 

Frictional  Resistance. 

1.  Frictional  resistance  =  R. 

8.  Force  to  overcome  resistance  =  FB  =  It  v. 

9.  Energy  expended  in  the  time  d  t  in  overcoming  resist- 

ance =  d  VrR=FRds  =  Rv*dt. 

Inertia. 

10.  Force  to  overcome  inertia  =  Ff  =  Ma  =  M-J--  . 

11.  Kinetic  energy  acquired  in  the  time  d  t 

dt 

12.  Kinetic  energy  =  W  =  f  Mv^dt  = 

Resistance  plus  Inertia. 

d  v 

13.  Total  force  applied  =  F  =  FR  +  F'  =  Rv  +  M^  . 

14.  Total  energy  supplied  in  the  time  dt 

or,     Fds  =  FRds  +  F'ds-y 


313 


(i   77 

or,     Fvdt  = 


314  APPENDIX  R 

TABLE  II.   EOTARY  MOTION. 

Notation. 

1.  Time  =  t. 

2.  Angle  =  0. 

3.  Angular  velocity  =  GO  =  -T-  ;     or,     d<p  =  aodt. 


dco 

4.  Angular  acceleration  =  a  =  —77-  =  -yrr  . 

5.  Moment    of    inertia    =  /. 

6.  Angular   momentum   = 

Frictional  Resistance. 

7.  Frictional  resistance  =  R. 

8.  Torque  to  overcome  resistance  =  TR  =  ROD. 

9.  Energy  expended  in  the  time  d  t  in  overcoming  resist- 

ance =  d  WR  =  TRd  0  =  R  tfd  t. 

Inertia. 

10.  Torque  to  overcome  inertia  —  T'  —  la  =  J-rr  • 

11.  Kinetic  energy  acquired  in  the  time  dt 

T'd</>=lGO~dt. 

12.  Kinetic  energy  =  W  =    f^Iao—dt  =  J  Ioo\ 

Resistance  plus  Inertia. 

13.  Total  torque  applied  =  T  =  TR+  T'  =  R  GO  +  7^. 

14.  Total  energy  supplied  in  the  time  d  t 

=  dW=dWB  +  dW';    or,     Tdcf>  =  TRd0+  T'd<f>; 


or,     Tcodt  =  £a>'dt  +  Ico-^dt. 


APPENDIX  B.  315 

TABLE  III.  —  ELECTRIC.  CURRENT. 

Notation. 

1.  Time  =  t. 

2.  Quantity  =  g. 

3.  Current  =  i  =  -~\     or,     dq  —  idt. 

4.  Current  acceleration  =  (3  =  -TT  . 

5.  .  Coefficient  of  self-induction  =  L. 

6.  Electro-magnetic  momentum  ==  Li. 

Ohmic  Resistance. 

7.  Ohmic  resistance  =  E. 

8.  Electromotive  force  to  overcome  resistance  =  eR  =  Hi. 

9.  Energy  expended  in  the  time  d  t  in  overcoming  resist- 

ance =  d  WR  =  eRdq  =  £  i*  d  t. 

/Self-induction. 
10.  Electromotive  force  to  overcome  self-induction 


11.  Energy  acquired  by  the  magnetic  field  in  the  time  d  t 


T         rfj 

12.  Energy  of  magnetic  field  =  W  =f  Lijjdt  = 

Resistance  plus  Self-induction. 

13.  Total  electromotive  force  applied 

=  e  =  eR  +  e'  =  E  i  +  Z^-. 

14.  Total  energy  supplied  in  the  time  d  t 

=  dW=dWR  +  dW;    or,     edq  =  eRdq  +  efdq  ; 

di 

or,     eidt  =  Ri*  dt  +  Li-^-dt. 


APPENDIX  C. 

NOTATION  USED  THROUGHOUT  THIS  BOOK. 

(Numbers  refer  to  page  where  first  used.) 

A.  Area,  67 ;  or,  constant,  41. 

B.  Constant,  41. 

B.  Induction  per  square  centimeter,  22. 

C.  Capacity,  64 ;  or,  constant,  41. 
C'.     Equivalent  capacity,  279. 

D.  Symbolic  operator,  84. 

E.  Constant  E.  M.  F.,  25  ;  or,  maximum  value  of  har- 

monic E.  M.  F.,  50. 

K    Virtual  E.  M.  F.,  i.e.,  square  root  of  mean  square 
value,  38  and  143. 

F.  Force,  20. 

H.     Magnetizing  force,  21. 

I.     Constant  current,  25 ;  or,  maximum  value   of   har- 
monic current,  53. 

/.     Virtual  current,  i.e.,   square   root   of   mean   square 

value,  38  and  143. 
Im.     Impedance,  188. 

Z.     Coefficient  of  self-induction,  23. 
Lf.     Equivalent  self-induction,  235. 

N.     Total  induction,  i.e.,  total  number  of  lines,  21. 

0.     Origin.     Center  of  revolution,  33. 

Q.     Constant  quantity  ;  or,  charge  of  electricity,  25. 

It.     Resistance,  24. 

R'.     Equivalent  resistance,  235. 

316 


APPENDIX  C.  317 


T.     Period,  33 ;  or,  time  constant,  46. 
V.     Potential,  63. 
W.     Work  or  energy,  28. 


a.  Amplitude,  33 ;  or,  constant,  86. 

b.  Constant,  57. 

c.  Arbitrary  constant  of  integration,  44. 

d.  Distance,  67. 

e.  Instantaneous  value  of  electromotive  force,  25. 
/.  Arbitrary  function,  43. 

/'.  First  differential  coefficient  of/,  71. 

h.  Constant,  184. 

i.  Instantaneous  value  of  current,  25. 

j.  v^~i,  93. 

L  Constant,  183. 

I.  Constant  length,  201. 

m.  Strength  of  pole,  20 ;  or,  constant,  96. 

n.  Frequency,  34  ;  or,  constant,  58. 

p.  An  abbreviation,  191. 

q.  Instantaneous  value  of  charge,  25. 

r.  Distance,  20;  or,  constant,  190. 

t.  Time,  34. 

x.  Independent  variable,  41 ;  also  length   or  distance, 

178 

y.  Dependent  variable,  34. 

3.  Dependent  variable,  42. 


a.  An  abbreviation,  191 ;  or,  a  constant,  41. 

ft.  Constant,  41. 

y.  Constant,  41. 

e.  Naperian  base,  (2.71828),  44. 

-f-  0.  Angle,  usually  of  advance,  35. 

—  9.  Angle,  usually  of  lag,  35. 


318  APPENDIX  G. 

K.  Specific  inductive  capacity,  61 ;  or,  Constant,  206, 

A.  Wave-length,  196. 

j*.  Permeability,  22. 

TT.  Eatio  of  circumference  to  diameter,  (3.14159),  21. 

2.  Summation,  59. 

r.  1  -r-  time- constant,  -^  ,  126. 

0.  Arbitrary  constant,  95. 

0.  Angle,  34. 

%.  Angle,  150. 

ip.  Current  angle,  55. 

co.  Angular  velocity,  2  n  n,  34. 


For  graphical  conventions,  see  219. 


INDEX. 


Acceleration,  unit  of,  18 

Addition  of  harmonic  electromotive 

forces.  213 

Addition  of  harmonic  functions,  38 
Advance,  angle  of,  35,  78,  134 
Air-chamber  analogue  of  condenser, 

272 

Ampere,  22 
Amplitude,  33 
Analogies,  mechanical,  313 
Analytical  treatment  (see  Contents), 

7,  17 

Angle  of  advance,  35,  78,  134 
Angle  of  lag,  35,  54,  134 
Angle  of  phase,  34 
Angle  of  epoch,  34 
Angular  velocity,  34,  50 
Apparent  resistance.  53,  79,  131 
Arrows,  meaning  of,  221 
Attraction,  law  of. 

—  for  charged  bodies,  60 

—  for  magnetic  poles,  20 
Average  value  of  sine-curve,  36 

B 

B,  Induction,  22 
Backward  waves,  202,  205 
Ballistic  galvanometer,  26 

C 

(,'able,  distributed  capacity  of,  176 
Capacity,  distributed,  176 
Capacity,  effects  of  variation  of, 

—  in  parallel  circuits  with  resist- 
ance, 285 

—  in  series  circuit  with  resistance, 
276 

—  in  series  circuit  with  resistance 
and  self-induction,  138,  298 


Capacity,  equivalent, 

—  for  parallel  circuits,  281,  283 

—  for  parallel  circuits  with  self- 
induction,  303 

—  for  series  circuits,  279 

—  for  series  circuit  with  self-in- 
duction, 296,  300 

Capacity  of  a  condenser,  65 
Capacity  of  a  conductor,  64 
Capacity  of  continuous  wire,  67 
Capacity  of  parallel  plates,  67 
Capacity  varied 

—  in  parallel  circuits  with  resist- 
ance, 285 

—  in  series  circuit  with  resistance, 
276 

—  in  series  circuit  with  resistance 
and  self-induction,  138,  298 

C.  G.  S.  units,  18,  312 
Charge,  energy  of,  64 
Charge  equation 

—  for  any  periodic  E.  M.  F.,  157 

—  for  charging  a  condenser,  118 

—  for  circuits  with  resistance  and 
capacity,  72 

—  for  circuits  with  resistance  and 
capacity  and  harmonic  E.  M.F. , 
78 

—  for  discharge  through  circuits 
with  resistance,  self-induction, 
and  capacity  : 

general  forms,  97 

nou -oscillatory,  99 

oscillatory,  107 

when  R2  0  =  4  L,  109 

—  for  discharging  condenser,  73 

—  for    non-oscillatory  charging, 
115 

—  for  oscillatory  charging,  120 

—  when  2P  C=  4  L,  122 

319 


320 


INDEX. 


Charge,  for  circuits  with  resistance, 
self-induction,  and  capacity,  112 
Charge,  general  solution   for, 

—  in  circuits  with  resistance  and 
capacity,  72 

—  in  circuits  with  resistance,  self- 
induction,  and  capacity,  87 

Charge  of  a  condenser,  74 
Charge,  unit  of,  25,  61 
Charging  a  condenser,  117 
Charging  equations,  three  forms  of, 

114 
Charging,  non-oscillatory,  114 

-  determination  of  constants,  114 

—  discussion,  116 
Charging,  oscillatory,  119 

—  determination  of  constants,  119 

—  discussion,  120 
Charging  when  R2  C  =  4L,  121 

—  determination  of  constants,  121 

—  discussion,  122 
Circles,  meaning  of,  221 
Closed  arrows,  meaning  of,  221 
Closed  circuits,  wave-propagation  in, 

Coefficient  of  self-induction,  23 
Combination  circuits 

—  with  resistance  and  capacity, 
286,  287,  288 

—  with  resistance  and  self-induc- 
tion, 248 

—  with  resistance,  self-induction, 
and  capacity,  310 

Complementary  function,  73,  92, 
96 

Composition  of  harmonic  electromo- 
tive forces,  213 

Composition  of  harmonic  functions, 
38 

Condenser.  65 

—  capacity  of,  65 

—  discharge  of,  72 

—  electromotive  force  of,  69 

—  energy  of,  65 

—  mechanical  analogue  of,  272 
Conductor,     energy     of,      when 

charged,  64 

Conjugate  imaginaries,  94 
Constant  current  example,  246 
Constant  potential  example,  245 
Constants,  variation  of, 

—  in  parallel  circuits  with  resist- 
ance and  capacity,  285 

—  in  parallel  circuits  with  resist- 
ance and  self-induction,  242 

—  in  series  circuit  with  resistance 
and  capacity,  274 

—  in  series  circuit  with  resistance 
and  self-induction,  222 


Constants,  variation  of,  in  series  cir- 
cuit with  resistance,  self-induc- 
tion, and  capacity,  134,  :296 

Construction  of  logarithmic  curve, 
46 

Continuous  conductor,  capacity  of. 

Conventions  adopted  219,  316 
Cosine  expanded,  93 
Cosine,  exponential  form  of,  93 
Coulomb's  law 

—  for  attraction  between  poles,  20 

—  for  attraction  between  charged 
bodies,  60 

Counter-electromotive  force  of  self- 
induction,  29 
Critical  case  of  discharge,  108 

—  of  charge,  121 
Criterion  of  iutegrabiliry,  43,  71 
Current  at  the  "  make,"  55,  144 
Current  equation, 

—  any  periodic  E.  M.  F.,  157 

—  for  charging  a  condenser,  118 

—  for    circuits  containing   resist- 
ance only,  133 

—  for  circuits  with  capacity  only, 
134 

—  for    circuits    with    distributed 
self-induction  and  capacity,  192 

—  for  circuit  with  resistance  and 
capacity,  72 

—  for  circuits  with  resistance  and 
capacity,     and     an     harmonic 
E.M.F.,  78,  133 

—  for  circuit  with  resistance  and 
self-induction,     and     harmonic 
E.  M.  F.,  53,  132 

—  for  discharging  condenser,  73 
Current    equation     for      discharge 

through   circuit   with  resistance, 
self-induction,  and  capacity,  92,  97 

—  nou-oscillatoiy.  99 

—  oscillatory,  107 

—  when  &  C  ^  4  L,  109 
Current  equation  for  establishment 

of  current  in  circuit  with  It  and 
L,  117 
Current  equation, 

—  for    non-oscillatory   charging, 
115 

—  for  oscillatory  charging,  120 

—  when  #2  C  -  4  L,  122 
Current,  general  solution  for, 

—  in  circuits  with  resistance  and' 
self-induction,  44 

—  in  circuits  with  resistance  and 
capacity,  72 

—  in  circuits  with  resistance,  self- 
induction,  and  capacity,  84,  86 


INDEX. 


321 


Current  graphically  shown  by  closed 

arrows,  221 
Current,  unit  of,  22 
Curves,  types  of,  163 

D 

D,  symbolic  operator,  84 
Decay  of  waves 

—  in  circuit  with  distributed  ca- 
pacity, 197 

—  in  circuit  with  distributed  ca- 
pacity and  self-induction,  200 

Decreasing  amplitude  of  waves 

—  in  circuits  with  distributed  ca- 
pacity, 197 

—  in  circuits  with  distributed  ca- 
pacity and  self-induction,  199 

Direction  of  rotation,  221 
Direction  of   rotation  of  E.  M.  F. 

vectors,  261 
Differential  equations 

—  for  charging,  113 

—  for  circuits  with  resistance  and 
capacity,  72 

—  for  circuits  with  resistance  and 
self-induction,  43 

—  for  circuits  with  resistance, self- 
induction,  and  capacity,  84 

—  for  discharge,  91 
Dimensions  of  impediment,  132 
Dimensions  of  L  GO,  55 
Discharge  through  circuit  with  re- 
sistance and  capacity,  72,  104 

Discharge  through  circuit  with  re- 
sistance, self-induction,  and  ca- 
pacity. 90 

—  non-oscillatory,  98 

—  oscillatory,  105 

—  when  R*  C=4L,  108 
Distributed  capacity,  176 

—  with  nc  self-induction,  194 

—  with  self-induction,  198 
Divided  circuit 

-with  resistance  and  capacity, 

280,  282,  284 

with  resistance  and  self-induc- 
tion, 233,  236,  241 

with  resistance,  self-induction, 

and  capacity,  302,  308,  310 
Dying  away  of    current  in  circuit 
with  resistance  and  self-induc- 
tion, 44,  103 
Dyne,  18 

E 

Et  <?,  Electromotive  force,  25 

Earth  inductor,  26 

Effective  electromotive  force,  55 


Effects  of  varying  constants 
-  in  parallel  circuits,  242,  285 

—  in  series  circuits,  134,  222,  274, 
296 

Electrical  analogies,  313 
Electrical  horse-power,  29 
Electromagnetic  induction,  21 
Electromotive  force 

—  diagram  for  circuits  with  re- 
sistance, self-induction,  and  ca- 
pacity, 293 

—  equation  for  circuit  with  resist- 
ance and  capacity,  69,  70 

—  equation  for  circuit  with  resist- 
ance and  self-induction,  31,  42 

—  equation  for  circuit  with  resist- 
ance,   self-induction,    and    ca- 
pacity, 83 

—  equation   for  circuit  with  dis- 
tributed self-induction  and  ca- 
pacity, 190 

—  graphically    shown    by    open 
arrows  221 

—  law  of,  23 

—  maximum  value  of,  50 

—  of  condenser,  69 

—  of  condenser  graphically  rep- 
resented, 269,  271 

—  of  self-induction,  29 

—  of    self-induction    graphically 
represented.  220 

—  triangle  of,   for  circuits  con- 
taining resistance  and  capacity, 
268 

—  triangle   of,  for  circuits  with 
resistance    and    self-induction, 
217 

—  unit  of,  24 
Electromotive  forces 

—  in  parallel,  263 

—  in  series,  260 

—  with  different  periods,  264 
E.  M.  F.  vectors,  rotation  of,  261 
Energy  dissipated  in  heat,  27 
Energy,  equation  of,  for  circuit  with 

resistance  and  capacity,  67 

—  equation   of,  for  circuits  with 
resistance  and  self-induction,  HO 

—  equation  of,  for  circuits  wiih 
resistance,    self-induction,    and 
capacity,  82 

—  imparted  to  a  circuit,  29,  142 
of  a  charged  conductor,  64 

—  of  a  condenser,  66 

—  of  magnetic  field,  29 

—  unit  of,  28 
Epoch,  34 

Equation  of  energy  for  circuits  with 
resistance  and  capacity,  67 


322 


INDEX. 


Equation  of  energy 
— for  circuits  with  resistance  and 
self-induction,  30 

—  for    circuits    with    resistance, 
self-induction,  and  capacity,  82 

Equation  of  E.  M.  F.'s 

—  for  circuits  with  resistance  and 
capacity,  69,  70 

—  for  circuits  with  resistance  and 
self-induction,  31 

—  for    circuits    with    resistance, 
self-induction,  and  capacity,  83 

Equivalent  capacity 

—  of  parallel  circuits,  281,  283 

—  of  parallel  circuits    with  self- 
induction,  303 

—  of  series  circuits,  279 

—  of  series  circuits  with  self-in- 
duction, 296,  300 

Equivalent  resistance 

—  of  parallel  circuits,  235,  238, 
281,  283,  303 

—  of  series  circuits,  279,  300 
Equivalent  self-induction 

—  of  parallel  circuits,  235,  238 

—  of    parallel   circuits  with    ca- 
pacity, 303 

—  of  series  circuits,  296,  300 
Erg,  unit  of  energy,  28 
Establishment  of  current 

—  in  circuit  with  resistance  and 
capacity,  117 

—  in  circuit  with  resistance  and 
self-induction,  48 

—  in  circuit  with  resistance,  self- 
induction,  and  capacity,  112 

Example  of  a  divided  circuit  with 
resistance,  self-induction,  and  ca- 
pacity, 309 

Expansion  of  sine  and  cosine,  93 
Expenditure  of  energy  in  a  circuit, 

82 

Explanation  of  exponential  term,  55 
Exponential  form   of   sine  and  co- 
sine, 93 
Exponential  term, 

—  effect  of ,  at  "make,"  56 

—  explanation  of,  55 


Faraday's  law,  23 
Field  of  force,  18 

—  intensity  of,  20 

—  unit  field,  21 
Force, 

—  law  of,  for  charged  bodies,  60 

—  law  of,  for  magnetic  poles,  20 

—  unit  of,  19 


Formulae  of  reduction,  51,  52 
Forward  waves,  202,  205 
Fourier's  Theorem,  41 
Frequency,  34 

—  variation  of,  140,  298 
Fundamental  units,  18 


Galvanometer,  ballistic,  26 
General  solution  for  charge 

—  in  circuits  with  resistance  and 
capacity.  72 

—  in  circuits  with  resistance,  self- 
induction,  and  capacity,  87 

General  solution  for  current 

—  in  circuits  with  resistance  and 
capacity,  72 

—  in  circuits  with  resistance  and 
self-induction,  44 

—  in  circuits  with  resistance,  self- 
induction,  and  capacity,  84,  86 

Graphical  representation 

-  of  a  simple  harmonic  E.M.F., 
212 

—  of  the  sum  of  several  harmonic 
E.  M.  F.'s,  213 

Graphical  treatment,  11.  209 

-  symbols  adopted,  219 


H 

H,  magnetizing  force,  21 
Harmonic  electromotive  force 

—  discussion,  130 

—  general  solution,  124 

—  graphical  representation  of,212 

—  in  circuit  with  resistance  and 
capacity,  76 

—  in  circuit  with  resistance  and 
seif-iuduction,  50 

—  solution  from  differential  equa- 
tions. 127 

Harmonic  functions,  32 

—  addition  of,  38 
Harmonic  motion,  33 
Heating  effect,  27 

—  same  with  as  without  self-in- 
duction and  capacity,  163 

Horse  power,  electrical,  29 


I 

/,  i,  current,  25 

I,  maximum  value  of  current,  53,  79, 

131 

Impedance,  53.  79,  131 
—  measurement  of,  230 


INDEX. 


323 


Impediment,  131,  295 
—  dimension  of,  132 
Impressed  electromotive  force,  55 
Induction,  21 
Inductive  resistance,  54 
Infinite  capacity,  67 
1  ntegrability,  criterion  of,  43,  71 
Integration  by  parts,  51 
Intensity  of  a  field  of  force,  20 


j  =  |/  _  i,  93 

Joule,  unit  of  energy,  28 

Joule's  law,  26 

Just  non-oscillatory 

—  charge,  121 

—  discharge,  110 

K 

K,  specific  inductive  capacity,  61 


L,  coefficient  of  self-induction,  23 
Lag, 

—  angle  of,  35,  54,  133,  134 

—  measurement  of,  230 
Law  of  attraction 

—  for  charged  bodies,  60 

—  for  magnetic  poles,  20 
Law  of  Coulomb,  20 

—  of  Faraday,  23 

—  of  Joule,  24 

—  of  Ohm,  26 

Linear  equation,  43,  44,  86 
Line  of  force,  18,  21 
Lines  of  induction,  21 
Limitations  of  the  telephone,  200 
Logarithmic  curve,  construction  of, 

46 
Loo,  dimensions  of,  55 

M 

Magnetic  field,  energy  of,  29 

—  intensity  of,  20 

Magnetic  pole,  18 

Magnetizing  force,  21 

Make,  current  at,  56,  144 

Maximum  oscillation,  153 

Maximum  value  of  harmonic  cur- 
rent, 5:},  79,  131 

Mean  square  value  of  a  sine-curve, 

37 
Measurement     by     three- voltmeter 

method,  230 
Mechanical  analogue  of  condenser, 

272 
Mechanical  analogies,  313 


Method  used  in  graphical  treatment, 

219 
Multiple-arc  arrangement 

—  of  circuits  with  resistance  and 
capacity,  289 

—  of  circuits  containing  resistance 
and  self-induction,  256 

Multiple-valued  function,  38 
ju,  permeability,  22 


N 

N,  total  induction,  21 
n,  frequency,  34 

Negative  direction  of  rotation,  221 
Neutralizing  of  self-induction  and 
capacity 

—  at  every  point  of  time,  158 

—  necessary  conditions  for,  162 
Neutralizing  of  self-induction  and 

capacity    impossible    except    for 
sine-curve,  175 
Non-oscillatory  charging,  114 

—  determination  of  constants,  114 
— discussion,  116 

Non-oscillatory  discharge,  98 

—  determination  of  constants,  98 

—  discussion,  99 

Notation,  219  (see  also  Appendix)  316 


O 

Ohm's  law,  24,  158 
Ohm,  unit  of  resistance,  24 
Open  arrows,  meaning  of,  221 
Oscillation  a  maximum,  153 
Oscillatory  charging,  119 

—  determination  of  constants,  119 

—  discussion,  120 
Oscillatory  discharge,  105 

—  determination  of  constants,  105 

—  discussion  of,  107 


Parabola  and  sine-curve  example, 

167 
Parallel  circuits 

—  with  resistance  and  capacity, 
280,  282,  284 

—  with  resistance  and  self-induc- 
tion, 233,  236,  241 

—  with  resistance,  self-induction, 
and  capacity,  302,  308,  310 

Parallel  plates,  capacity  of,  67 
Particular  E  M.  F.'s,  87 
Period,  33 
Periodic  functions,  38 


324 


INDEX. 


Periodic  E.  M.  F.  in  circuit  with 
resistance  and  capacity,  79 

—  in  circuit  with  resistance  and 
self-induction,  57 

—  in  circuit  with  resistance,  self- 
induction  and  capacity,  124 

Periodicity,  34 

Permeability,  22 

Phase,  34 

Pole,  unit  magnetic,  18 

Positive  and  negative  flow  of  alter- 
nating current  equal,  164 

Positive  direction  of  rotation,  33,  221 

Potential,  61,  63 

Potential  of  a  conductor  w7ith  dis- 
tributed self-induction  and  ca- 
pacity, 190 

Power,  measurement  of,  by  three 
voltmeters,  232 

Practical  units,  312 

Problems,  see  Contents,  11 

Propagation  of  waves,  rate  of, 

—  in    circuits    with     distributed 
capacity,  195 

—  in    circuits    with    distributed 
capacity  and  self-induction,  198 

Q 

Quantity,  Q, 

—  definition  of.  25 

—  for  half  period,  164 

—  unit  of,  25,  61 
Quickest  charge.  121 
Quickest  discharge,  110 

R 

Rate  of  decay  of  waves 

—  in  circuit  with  distributed  ca- 
pacity, 197 

—  in  circuit  with  distributed  capa- 
city and  self-induction.  200 

Rate  of  propagation  of  waves 

—  in  circuit  with  distributed  ca- 
pacity, 195 

—  in  circuit  with  distributed  ca- 
pacity and  self-induction,  198 

Rate  of  work,  28 

Resistance,  effect  of  variation  of, 

—  in    parallel    circuit    with    ca- 
pacity, 285 

—  in  parallel  circuit  with  self-in- 
duction, 242 

—  in  series  circuit  with  capacity, 
274 

—  in   series   circuit  with  self-in- 
duction, 223 

—  in  series  circuit  with  resistance, 
self-induction,  and  capacity,  135 


Resistance  equivalent, 

—  of  parallel,  circuits,  235,  238, 
281,  283,  303 

—  of  series  circuits,  279,  200 
Resistance,  R,  unit  of,  24 
Resultant     of     several     harmonic 

E.  M.  F.'s   of  the  same  period, 

213 

Rotation,  direction  of,  221 
Rotation  of  E.  M.  F.  vectors,  261 


Self-induction,  coefficient  of,  23 

— electromotive  force  of,  220 
Self-induction,  effect  of  variation  of, 

—  in  parallel  circuits,  244 

—  in  series  circuit,  225 

—  in  series  circuit  with  resistance, 
self-induction,     and     capacity, 
137,  298 

Self-induction,  equivalent, 

—  of  parallel  circuits,  235,  238 

—  of    parallel    circuits  with   ca- 
pacity, 303 

—  of  series  circuit  with  capacity, 
296,  300 

Self-induction,  measurement  of,  230 
Series  and  parallel  circuits 

—  with  resistance  and  capacity, 
286,  287,  288 

—  with  resistance  and  self-induc- 
tion, 248,  250,  251,  252 

—  with  resistance,  self-induction, 
and  capacit}7,  310 

Series  circuit 

—  with  resistance  and  capacity, 
278,  279 

—  with  resistance  and  self-induc- 
tion, 227,  228,  229 

—  with  resistance,  self-induction, 
and  capacity,  299,  301 

Several  sources  of  E.  M.  F.,  260 
Sine-curve,  35 

—  average  value  of,  36 

—  mean  square  value  of,  37 
Sine-curve   and  parabola  example, 

167 

Sine  expanded,  93 
Sine,  exponential  form  of,  93,  186 
Sine-functions,  32 
Single- value  function,  88 
Specific  inductive  capacity,  61 
Strength  of  a  magnetic  field,  20 
Sum  of  harmonic  E.  M.  F.'s  of  the 

same  period,  213 

Symbols  adopted  in  graphical  treat- 
ment, 219 
Symbolic  operator,  84,  128 


INDEX. 


325 


Telephone,  limitations  of  the,  200 

T,  period,  33 

T,  see  Time-constant. 

Three- voltmeter  method,  230 

Time-constant 

—  in  circuit  with  resistance  and 
capacity,  74 

—  in  circuit  with  resistance  and 
self-induction,  46 

—  in  circuits  with  resistance,  self- 
induction,  and  capacity,  85 

Transformation  to  real  form,  93 
Triangle  of  E.  M.  F.'s 

—  for  circuits  with  resistance  and 
capacity,  268 

—  for  circuits  with  resistance  and 
self-induction,  217 

—  for  circuits  with  resistance,  self- 
induction,  and  capacity,  293 

Two  E.  M.  F.'s  in  series,  260,  264 
Types  of  curves,  163 

U 

Unit  charge,  61 
Unit  current,  22 
Unit  magnetic  pole,  18 
Unit  of  energy,  28 
Unit  quantity,  25 


Variation  of  capacity 

—  in  parallel  circuits,  285 

—  in  series  circuits,  138,  276,  296 
Variation   of  constants  in  parallel 

—  circuits,  285 


Variation  of  constants  in  series  cir- 

—  cults,  134,  274,  296 
Variation  of  frequency 

—  in  series  circuits,  140,  296 
Variation  of  resistance 

—  in  parallel  circuits,  242,  285 

—  in  series  circuits,  135,  223,  274, 
296 

Variation  of  self-induction 

—  in  parallel  circuits,  244 

—  in  series  circuits,  137,  225,  296 
Velocity,  unit  of,  18 

Virtual  values  of  E.  M.  F.  and  cur- 
rent, 38,  54,  131,  143 
Volt,  24 

W 

W,  work  or  energy,  28 
Watt,  unit  of  work,  28 
Wave-length,  206 

Wave-propagation   in  circuits  with 
distributed  capacity, 

—  decreasing  amplitude  of,  197 

—  nature  of,  194 

—  rate  of,  195 
Wave-propagation  in  circuits  with 

distributed  capacity  and  self-in- 
duction, 

—  decreasing  amplitude  of,  199 

—  nature  of,  198 

—  rate  of,  198 

Wave- propagation  in  closed  circuits, 

201 
Work  done  by  harmonic  current, 

142 

Work  in  moving  charge,  62 
GO,  angular  velocity,  34 


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